Chapter 6 Visualising The Solid Shapes

A prism is a polyhedron characterised by two parallel and congruent polygonal bases.

The faces connecting the corresponding sides of the two bases are the lateral faces.

These lateral faces are always parallelograms.

In a right prism, the lateral faces are rectangles, which are a special type of parallelogram.

Therefore, the lateral faces of a prism are parallelograms.

The correct option is (c) Parallelograms.

A pyramid is a polyhedron formed by connecting a polygonal base to a common point, called the apex.

The faces formed by connecting the sides of the base to the apex are the lateral faces.

Each side of the polygonal base serves as the base of a triangle, and the apex is the common vertex for all these triangles.

Therefore, the lateral faces of a pyramid are always triangles.

The correct option is (b) Triangles.

A polyhedron is considered regular if its faces are all congruent regular polygons and the same number of faces meet at each vertex.

This property, where the same number of faces meet at each vertex, is a defining characteristic of regular polyhedra (along with congruent regular faces).

Therefore, the blank should be filled with the phrase describing this property.

In a regular polyhedron the same number of faces meet at each vertex.

A pentagonal prism has a pentagon as its base. A pentagon has 5 vertices and 5 edges.

A prism has two identical bases and lateral faces connecting corresponding sides.

Number of vertices in a pentagonal prism = Number of vertices in base $\times$ 2 = $5 \times 2 = 10$.

Number of faces in a pentagonal prism = Number of base faces + Number of lateral faces = $2 + 5 = 7$. (2 pentagonal bases and 5 parallelogram lateral faces).

To find the number of edges, we can count the edges on the bases and the lateral edges.

Edges on the two bases = Edges per base $\times$ 2 = $5 \times 2 = 10$.

Lateral edges (connecting the vertices of the two bases) = Number of vertices per base = $5$.

Total number of edges = Edges on bases + Lateral edges = $10 + 5 = 15$.

Alternatively, using Euler's formula for polyhedra: $V - E + F = 2$, where $V$ is vertices, $E$ is edges, and $F$ is faces.

We have $V=10$ and $F=7$.

Substituting these values: $10 - E + 7 = 2$

$17 - E = 2$

$E = 17 - 2 = 15$.

A pentagonal prism has 15 edges.

A polyhedron is a three-dimensional solid object whose surface is made up of a finite number of polygonal faces.

Key characteristics of a polyhedron include having flat faces, straight edges, and vertices (corners).

A sphere is a perfectly round geometrical object in three-dimensional space.

The surface of a sphere is curved; it does not have any flat faces, straight edges, or vertices.

Since a sphere lacks the fundamental defining characteristics of a polyhedron (flat faces, straight edges, vertices), it is not a polyhedron.

The statement "A sphere is a polyhedron" is False.

A prism is a polyhedron with two identical and parallel polygonal bases and lateral faces that are parallelograms.

The lateral faces connect corresponding sides of the two bases.

If the base polygon is irregular (i.e., its sides are not all equal in length), then the corresponding sides in the two bases will also have different lengths.

The lateral faces are parallelograms whose width is the length of the corresponding side of the base and whose height is the perpendicular distance between the bases (for a right prism) or the length of the lateral edge (for an oblique prism).

If the base has sides of different lengths, the lateral faces will be parallelograms with different base lengths, and thus they will not be congruent.

For example, a right prism with a rectangular base that is not a square will have two pairs of congruent rectangular lateral faces, but the two pairs will not be congruent to each other if the rectangle's length and width are different.

Therefore, it is true that the lateral faces of a prism need not be congruent.

The statement "In a prism the lateral faces need not be congruent" is True.

The top, front, and side views of the given solid represent how the solid appears when viewed directly from above, from the front, and from one of the sides, respectively.

Front View:

Looking at the solid from the front, we observe a stepped structure. This view will show three rectangular areas stacked vertically, with each upper rectangle set back or forward relative to the one below it, creating a staircase profile.

Top View:

Looking down at the solid from above, we see the extent of the solid on the horizontal plane. This view will show the rectangular base and the outlines of the upper layers. It will appear as a set of concentric or aligned rectangles, showing the decreasing area covered at each successive level.

Side View:

Looking at the solid from the side (assuming the side perpendicular to the front view), we will see the overall rectangular outline of the solid from that perspective. The edges formed by the steps on the front view will appear as vertical lines within this rectangle in the side view, dividing it into vertical sections corresponding to the different depths of the steps.

Isometric dot paper provides a grid of dots arranged in equilateral triangles, which helps in drawing 3D objects in isometric projection. To sketch a rectangular prism, we need to represent its length, width, and height along the three isometric axes.

Follow these steps:

1. Choose a starting dot on the isometric paper. This will represent one vertex of the rectangular prism (e.g., a corner of the base).

2. From this starting dot, draw a line segment of length 4 units along one of the isometric grid directions (e.g., following the dots diagonally upwards and to the right). This represents the length of the prism.

3. From the same starting dot, draw a line segment of width 3 units along another isometric grid direction (e.g., following the dots diagonally upwards and to the left). This represents the width of the prism.

4. From the end point of the 4-unit line, draw a line segment of 3 units parallel to the width line drawn in step 3. From the end point of the 3-unit line, draw a line segment of 4 units parallel to the length line drawn in step 2. These two lines should meet, completing the base parallelogram of the prism.

5. From each of the four vertices of the base parallelogram, draw a vertical line segment upwards, each of height 2 units (following the vertical grid lines). This represents the height of the prism.

6. Connect the top end points of these vertical line segments. The resulting figure at the top will be a parallelogram congruent to the base parallelogram. The edges connecting the top vertices should be parallel to the corresponding base edges.

7. For clarity, you can draw the visible edges with solid lines and the hidden edges (usually the three edges on the back side) with dashed lines.

The resulting sketch on the isometric dot paper will be an isometric view of the rectangular prism with the specified dimensions.

The given figure is a net, which is a 2D pattern that can be folded to form a 3D object.

Let's examine the components of the net.

The net consists of six congruent square faces arranged in a specific pattern.

When this net is folded along the edges, the squares will form the faces of a three-dimensional shape.

A polyhedron with six congruent square faces is known as a cube.

The shape whose net is given is a cube.

A rectangular prism, or cuboid, is a polyhedron with 6 rectangular faces, 12 edges, and 8 vertices.

Diagonals in a cuboid are line segments connecting two vertices that are not on the same face.

There are two types of diagonals in a cuboid: face diagonals and space diagonals.

Face Diagonals:

These are the diagonals drawn on each of the rectangular faces. Each rectangular face has two diagonals.

A cuboid has 6 faces.

Number of face diagonals = Number of faces $\times$ Diagonals per face

Number of face diagonals = $6 \times 2 = 12$.

On the given figure, these would be lines like AC, BD (on the bottom face), EG, FH (on the top face), AB', DC' (on one side face), AD', BC' (on another side face), AE, BF (on the front face), CG, DH (on the back face), where B', C', D' represent the vertices on the back or side not labelled in the diagram, or using the likely vertex labels A, B, C, D for the base and E, F, G, H for the top (with E above A, F above B, etc.), the face diagonals are AC, BD, EG, FH, AF, BE, CG, DH, AH, DE, BG, CF.

Space Diagonals:

These are the diagonals that pass through the interior of the cuboid, connecting opposite vertices.

There are 4 space diagonals in a cuboid.

These connect vertex pairs (A, G), (B, H), (C, E), (D, F) based on the standard labeling where A,B,C,D form the base and E,F,G,H form the top corresponding to A,B,C,D respectively.

On the given figure (assuming A, B, C, D are bottom vertices and E, F, G, H are top vertices, e.g., A is bottom left front, B is bottom right front, D is bottom left back, C is bottom right back, and E, F, H, G are top corresponding vertices), the space diagonals are AG, BH, CE, DF.

Total Diagonals:

Total number of diagonals = Number of face diagonals + Number of space diagonals

Total number of diagonals = $12 + 4 = 16$.

To make (draw) all the diagonals on the shape, one would draw these 16 line segments connecting the appropriate pairs of vertices.

To count the number of cubes in the given shapes, we can examine the structures layer by layer from the base or count the cubes in each visible column.

Shape 1 (Left Structure):

Let's count layer by layer from the bottom.

The bottom layer appears to be a complete $3 \times 3$ square arrangement.

Number of cubes in Layer 1 = $3 \times 3 = 9$.

The second layer sits on top of the central $2 \times 2$ section of the first layer.

Number of cubes in Layer 2 = $2 \times 2 = 4$.

The third layer sits on top of the central $1 \times 1$ section of the second layer.

Number of cubes in Layer 3 = $1 \times 1 = 1$.

Total number of cubes in Shape 1 = Sum of cubes in all layers = $9 + 4 + 1 = 14$.

Shape 2 (Right Structure):

Let's count the height of the stacks in each position of the $2 \times 2$ base.

From the viewing angle, we can see four vertical stacks (columns).

The front-left stack has a height of 3 cubes.

The front-right stack has a height of 3 cubes.

The back-left stack has a height of 3 cubes.

The back-right stack has a height of 1 cube.

Total number of cubes in Shape 2 = Sum of cubes in all stacks = $3 + 3 + 3 + 1 = 10$.

Number of cubes in Shape 1 is 14.

Number of cubes in Shape 2 is 10.

We need to identify each polyhedron and then count its vertices ($V$), edges ($E$), and faces ($F$) to verify Euler's formula $V - E + F = 2$.

Polyhedron (i):

This polyhedron has a triangular base and three triangular lateral faces that meet at a single apex.

This shape is a triangular pyramid (also known as a tetrahedron).

Let's count the vertices, edges, and faces:

Vertices ($V$): There are 3 vertices on the base and 1 apex vertex. So, $V = 3 + 1 = 4$.

Edges ($E$): There are 3 edges on the base and 3 lateral edges connecting the base vertices to the apex. So, $E = 3 + 3 = 6$.

Faces ($F$): There is 1 triangular base face and 3 triangular lateral faces. So, $F = 1 + 3 = 4$.

Now, let's verify Euler's formula $V - E + F = 2$:

$V - E + F = 4 - 6 + 4 = 8 - 6 = 2$.

Since $V - E + F = 2$, Euler's formula is verified for the triangular pyramid.

Polyhedron (ii):

This polyhedron has two identical rectangular bases that are parallel and congruent. The lateral faces are parallelograms (or rectangles in this case, appearing as a right prism).

This shape is a rectangular prism (also known as a cuboid).

Let's count the vertices, edges, and faces:

Vertices ($V$): There are 4 vertices on the bottom base and 4 corresponding vertices on the top base. So, $V = 4 + 4 = 8$.

Edges ($E$): There are 4 edges on the bottom base, 4 edges on the top base, and 4 vertical lateral edges connecting the bases. So, $E = 4 + 4 + 4 = 12$.

Faces ($F$): There are 2 rectangular base faces (bottom and top) and 4 rectangular lateral faces. So, $F = 2 + 4 = 6$.

Now, let's verify Euler's formula $V - E + F = 2$:

$V - E + F = 8 - 12 + 6 = 14 - 12 = 2$.

Since $V - E + F = 2$, Euler's formula is verified for the rectangular prism.

We are given the number of faces ($F$) and the number of vertices ($V$) of a polyhedron, and we need to find the number of edges ($E$).

We can use Euler's formula for polyhedra, which relates the number of vertices, edges, and faces:

$\qquad V - E + F = 2$

Given:

Number of faces, $F = 7$

Number of vertices, $V = 10$

We need to find $E$.

Substitute the given values into Euler's formula:

Combine the constant terms on the left side:

$17 - E = 2$

To find $E$, rearrange the equation:

$E = 17 - 2$

Calculate the value of $E$:

$E = 15$

The polyhedron has 15 edges.

We are given the number of edges ($E$) and the number of faces ($F$) of a polyhedron, and we need to find the number of vertices ($V$).

We can use Euler's formula for polyhedra, which relates the number of vertices, edges, and faces:

$\qquad V - E + F = 2$

Given:

Number of edges, $E = 30$

Number of faces, $F = 12$

We need to find $V$.

Substitute the given values into Euler's formula:

Simplify the left side of the equation:

$V - 18 = 2$

To find $V$, add 18 to both sides of the equation:

$V = 2 + 18$

Calculate the value of $V$:

$V = 20$

The polyhedron has 20 vertices.

Given:

Distance between City A and City B on the map = $6$ cm.

Scale of the map: $1$ cm on the map represents $200$ km in reality.

To Find:

The actual distance between City A and City B.

Solution:

The scale of the map provides a ratio that relates a distance measured on the map to the corresponding actual distance on the ground.

The given scale is $1$ cm = $200$ km.

This means that every $1$ cm measured on the map corresponds to an actual distance of $200$ km.

The distance between City A and City B on the map is given as $6$ cm.

To find the actual distance, we multiply the distance on the map by the value that $1$ cm represents in reality.

Actual distance = Distance on map $\times$ Value of $1$ cm in reality

Actual distance = $6$ cm $\times$ $200$ km/cm

Actual distance = $6 \times 200$ km

$6 \times 200 = 1200$

Actual distance = $1200$ km

The actual distance between City A and City B is 1200 km.

Given:

Actual height of the building = 9 m.

Height of the building on the map = 9 cm.

To Find:

The scale used for the map.

Solution:

The scale of a map is the ratio of a distance on the map to the corresponding actual distance on the ground.

We are given that 9 cm on the map represents an actual distance of 9 m.

First, let's express the actual distance in the same units as the map distance. We convert meters to centimeters.

We know that 1 meter = 100 centimeters.

So, 9 meters = $9 \times 100$ cm = 900 cm.

Now we have the relationship:

9 cm on the map represents 900 cm in reality.

To find the scale as what 1 cm on the map represents, we divide both sides of this relationship by 9.

$\frac{9 \text{ cm}}{9} = \frac{900 \text{ cm}}{9}$

$1 \text{ cm} = 100 \text{ cm}$

Alternatively, since 100 cm equals 1 meter, the scale can also be expressed as:

$1 \text{ cm} = 1 \text{ m}$

The scale used for the map is 1 cm = 1 m or 1:100.

Given:

Scale of the map: 1 mm represents 4 m.

Actual distance = 52 m.

To Find:

The distance on the map corresponding to the actual distance of 52 m.

Solution:

The scale tells us that:

4 meters in reality are represented by 1 mm on the map.

We want to find the map distance for an actual distance of 52 meters.

From the scale, we can find what 1 meter in reality is represented by on the map:

$4 \text{ m} \iff 1 \text{ mm}$

$\frac{4 \text{ m}}{4} \iff \frac{1 \text{ mm}}{4}$

$1 \text{ m} \iff \frac{1}{4} \text{ mm}$

Now, to find the map distance for 52 meters, we multiply the value for 1 meter by 52:

$52 \text{ m} \iff 52 \times \frac{1}{4} \text{ mm}$

Calculate the product:

$52 \times \frac{1}{4} = \frac{52}{4} = 13$

So, 52 meters in reality are represented by 13 mm on the map.

The distance on the map for an actual distance of 52 m is 13 mm.

The given figure is a polyhedron.

It has a hexagonal base and triangular faces meeting at an apex.

This shape is a hexagonal pyramid.

We need to determine the number of vertices, edges, and faces.

Vertices ($V$):

The base is a hexagon, which has 6 vertices.

There is one apex vertex above the base.

Total number of vertices = Vertices in base + Apex vertex

$V = 6 + 1 = 7$

Edges ($E$):

The base is a hexagon, which has 6 edges.

There are edges connecting each vertex of the base to the apex. Since there are 6 base vertices, there are 6 lateral edges.

Total number of edges = Edges in base + Lateral edges

$E = 6 + 6 = 12$

Faces ($F$):

There is one base face, which is a hexagon.

There are lateral faces connecting each edge of the base to the apex. Since there are 6 edges in the base, there are 6 triangular lateral faces.

Total number of faces = Base face + Lateral faces

$F = 1 + 6 = 7$

Summary:

Number of vertices, $V = 7$

Number of edges, $E = 12$

Number of faces, $F = 7$

We can verify Euler's formula ($V - E + F = 2$) for this polyhedron:

$V - E + F = 7 - 12 + 7$

$V - E + F = 14 - 12$

$V - E + F = 2$

The formula holds true.

Explanation:

A polyhedron is a three-dimensional solid figure whose surface is made up of a finite number of plane faces that are polygons.

Let's examine the given options:

(A) Cube: A cube is made up of 6 square faces. All faces are polygons. Therefore, a cube is a polyhedron.

(B) Pyramid: A pyramid is made up of a polygonal base and triangular faces that meet at a common vertex. All faces are polygons. Therefore, a pyramid is a polyhedron.

(C) Cone: A cone has a circular base and a curved lateral surface that tapers to a point (apex). It has a curved surface, which is not a polygon. Therefore, a cone is not a polyhedron.

(D) Prism: A prism is made up of two identical and parallel polygonal bases and rectangular faces connecting corresponding sides of the bases. All faces are polygons. Therefore, a prism is a polyhedron.

Based on the definition of a polyhedron, the cone is the figure that does not meet the criteria as it has a curved surface.

The correct option is (C) Cone.

Explanation:

A polyhedron is a three-dimensional solid whose surface is composed of a finite number of polygonal faces.

Let's consider each option:

(a) 3 triangles: It is not possible to form a closed three-dimensional figure with just three triangular faces. A tetrahedron, which is the simplest polyhedron, requires a minimum of four triangular faces.

(b) 2 triangles and 3 parallelograms: This combination can form a triangular prism. A triangular prism has two parallel triangular bases and three rectangular (or parallelogram) lateral faces. This is a polyhedron.

(c) 8 triangles: This combination can form an octahedron, which is a polyhedron with 8 triangular faces. It can also form other polyhedra like a pyramid with an octagonal base (which would have 8 triangular faces and 1 octagonal base, total 9 faces). Since it's possible to form a polyhedron, this option is valid for forming one.

(d) 1 pentagon and 5 triangles: This combination can form a pentagonal pyramid. It has a pentagonal base and 5 triangular faces meeting at an apex. This is a polyhedron.

Only option (a) cannot form a polyhedron as a closed three-dimensional shape requires a minimum of four faces.

The correct option is (a) 3 triangles.

Explanation:

A regular polyhedron is a polyhedron whose faces are congruent regular polygons and where the same number of faces meet at each vertex.

Let's examine each option:

(a) Cuboid: A cuboid has rectangular faces. While opposite faces are congruent, not all faces are necessarily congruent to each other (unless it's a cube). Rectangles are regular polygons only if they are squares. If the faces are not all congruent regular polygons, it is not a regular polyhedron.

(b) Triangular prism: A triangular prism has two triangular bases and three parallelogram (usually rectangular) lateral faces. The faces are not all congruent, and they are of different types (triangles and parallelograms). Therefore, it is not a regular polyhedron.

(c) Cube: A cube has six congruent square faces. A square is a regular polygon. At each vertex of a cube, three faces meet, and three edges meet. All faces are congruent regular polygons, and the same number of faces meet at each vertex. Therefore, a cube is a regular polyhedron.

(d) Square prism: A square prism has two square bases and four rectangular lateral faces. Unless the rectangular faces are also squares (making it a cube), the faces are not all congruent or of the same type. Therefore, it is generally not a regular polyhedron.

Among the given options, only the cube satisfies the conditions of a regular polyhedron.

The correct option is (c) Cube.

Explanation:

A two-dimensional (2D) figure is a flat shape that has only two dimensions: length and width. It lies entirely in a plane.

A three-dimensional (3D) figure is a solid object that has three dimensions: length, width, and height (or depth). It occupies space.

Let's examine each option:

(a) Rectangle: A rectangle is a flat shape with four straight sides and four right angles. It has length and width, and exists in a plane. Therefore, a rectangle is a two-dimensional figure.

(b) Rectangular Prism: A rectangular prism is a solid shape with six rectangular faces. It has length, width, and height. It occupies space. Therefore, a rectangular prism is a three-dimensional figure.

(c) Square Pyramid: A square pyramid is a solid shape with a square base and four triangular faces that meet at a point. It has length, width, and height. It occupies space. Therefore, a square pyramid is a three-dimensional figure.

(d) Square Prism: A square prism is a solid shape with two square bases and four rectangular lateral faces. It has length, width, and height. It occupies space. Therefore, a square prism is a three-dimensional figure.

Among the given options, only the rectangle is a two-dimensional figure.

The correct option is (a) Rectangle.

Explanation:

A pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. The lateral faces of a pyramid are triangles.

The base of a pyramid must be a polygon.

Let's examine each option to see if it is a polygon:

(a) Line segment: A line segment is a part of a line. It is a 1-dimensional object and not a polygon.

(b) Circle: A circle is a 2-dimensional closed curve. It is not made up of straight line segments, so it is not a polygon. A solid with a circular base and an apex is called a cone, which is not a pyramid (as it's not a polyhedron).

(c) Octagon: An octagon is an 8-sided polygon. A pyramid with an octagonal base is called an octagonal pyramid, and it is a valid type of pyramid.

(d) Oval: An oval is a 2-dimensional closed curve. It is not made up of straight line segments, so it is not a polygon.

Since the base of a pyramid must be a polygon, and an octagon is a polygon, an octagon can be the base of a pyramid.

The correct option is (c) Octagon.

Explanation:

A vertex (plural: vertices) is a point where edges or faces of a three-dimensional solid meet.

Let's examine the vertices of each given 3D shape:

(a) Pyramid: A pyramid has vertices. For example, a square pyramid has vertices at the corners of its square base and one vertex at the apex where the triangular faces meet.

(b) Prism: A prism has vertices. For example, a triangular prism has vertices at the corners of its triangular bases.

(c) Cone: A cone has one vertex at its apex, where the curved surface meets.

(d) Sphere: A sphere is a perfectly round shape in three-dimensional space. It has a continuous curved surface and no edges or corners where faces meet. Therefore, a sphere does not have any vertices.

Among the given options, only the sphere does not have a vertex.

The correct option is (d) Sphere.

Explanation:

Let's examine the edges of the given shapes:

(a) Polyhedron: A polyhedron is a three-dimensional solid bounded by polygonal faces. The intersection of two faces of a polyhedron forms an edge, and these edges are always line segments. Examples of polyhedra include cubes, prisms, and pyramids.

(b) Cone: A cone has a circular base and a curved lateral surface meeting at an apex. It has a single edge where the base meets the curved surface, which is a circle (a curved line), not a line segment.

(c) Cylinder: A cylinder has two parallel circular bases and a curved lateral surface. It has two edges where the bases meet the curved surface, and these are circles (curved lines), not line segments.

(d) Polygon: A polygon is a two-dimensional closed shape made up of line segments. The question asks for a solid (a 3D shape), so this option is incorrect as it is a 2D figure.

The solid shape among the options that has only line segments as its edges is a polyhedron.

The correct option is (a) Polyhedron.

Explanation:

We are given a solid with the number of faces ($F$) and the number of vertices ($V$). We need to find the number of edges ($E$).

Given:

We can use Euler's formula for polyhedra, which states the relationship between the number of faces, vertices, and edges:

Substitute the given values of $F$ and $V$ into Euler's formula:

Simplify the equation:

To find $E$, rearrange the equation:

So, the number of edges in this solid shape is 8.

A solid with F=5, V=5, and E=8 is a square pyramid (1 square base + 4 triangular faces = 5 faces; 4 base vertices + 1 apex vertex = 5 vertices; 4 base edges + 4 lateral edges = 8 edges).

The correct option is (c) 8.

Explanation:

We need to determine the shape seen when looking down directly from above the given 3D object.

The given 3D shape consists of two parts:

1. A larger rectangular block at the bottom.

2. A smaller square block placed centrally on top of the rectangular block.

When viewed from the top:

The bottom rectangular block will appear as its base rectangle.

The top square block will appear as its base square.

Since the square block is placed on top of the rectangular block, the top view will show the outline of the rectangular base with the outline of the square top block inside it. This is because the square is within the boundaries of the rectangle when viewed from above.

Therefore, the top view will be a large rectangle with a smaller square drawn inside it.

Now let's look at the given options:

Option (a) shows a single rectangle.

Option (b) shows a rectangle with a square inside it.

Option (c) shows a single square.

Option (d) shows a rectangle divided into smaller rectangles.

Comparing our expected top view with the options, option (b) matches the description of a rectangle with a square inside.

The correct option is (b).

Explanation:

We are given the net of a cube and asked to find the face opposite to the face marked with the letter L.

When folding a cube net, opposite faces are generally separated by one face in a straight line (either horizontally or vertically).

Let's visualize folding the given net:

The face marked L is in a horizontal strip with faces M, N, O, and P.

Starting from L and moving horizontally:

L is next to M.

M is next to N.

N is next to O.

O is next to P.

In a linear arrangement of faces that form a side or base of the cube, faces that are two steps apart will be opposite to each other when folded.

Looking at the horizontal strip: L, M, N, O, P

L is two steps away from N (L -> M -> N). So, L is opposite to N.

M is two steps away from O (M -> N -> O). So, M is opposite to O.

The face Q is attached to the face N. When the net is folded, Q will become the top or bottom face, and N will be one of the side faces.

Let's confirm using the rule that opposite faces are separated by one face in a line.

In the horizontal strip L-M-N-O-P:

L is opposite to N.

M is opposite to O.

The remaining face P is opposite to the face Q (which is connected to N and will be the other base/top face).

Therefore, the face opposite to the face marked with the letter L is the face marked with the letter N.

The correct option is (b) N.

Explanation:

A cone is a three-dimensional geometric shape that tapers smoothly from a flat, circular base to a point called the apex or vertex.

The net of a solid is a two-dimensional shape that can be folded to form the three-dimensional solid.

Let's analyze the nets provided in the images:

Net (a): This net consists of a triangle and a circle. When folded, the triangle will likely form a lateral face, and the circle will form a base. This type of net typically forms a pyramid with a triangular base, not a cone.

Net (b): This net consists of a sector of a circle and a circle. When the sector of a circle is rolled up, the two straight edges meet to form the curved surface of the cone, with the common point becoming the apex. The arc of the sector becomes the circumference of the base. The separate circle in the net forms the base of the cone. This net correctly represents the surface area of a cone and can be folded to form a cone.

Net (c): This net consists of two circles connected by a rectangle. When folded, the rectangle rolls up to form the curved lateral surface, and the two circles form the top and bottom bases. This net forms a cylinder, not a cone.

Net (d): This net consists of a polygon (in this case, a rectangle) and several triangles attached to its sides. When folded, the polygon forms the base, and the triangles form the lateral faces that meet at an apex. This net forms a pyramid with a rectangular base, not a cone.

Based on the structure of a cone and its net, the net that generates a cone is the one composed of a sector of a circle and a circle.

The correct option is (b).

Explanation:

A prism is a polyhedron consisting of a polygonal base, a translated copy of the base, and faces joining corresponding sides. The lateral faces of a prism are parallelograms (often rectangles in right prisms).

Let's examine each figure:

Figure (a): This shape has two parallel and congruent triangular bases and three rectangular lateral faces connecting the corresponding sides of the bases. This is a triangular prism.

Figure (b): This shape has a rectangular base and four triangular faces meeting at a single point (apex). This shape is a rectangular pyramid. The lateral faces are triangles, not parallelograms, and there is only one base, not two parallel congruent bases.

Figure (c): This shape has two parallel and congruent pentagonal bases and five rectangular lateral faces connecting the corresponding sides of the bases. This is a pentagonal prism.

Figure (d): This shape has two parallel and congruent hexagonal bases and six rectangular lateral faces connecting the corresponding sides of the bases. This is a hexagonal prism.

Based on the definition, a prism must have two parallel and congruent bases and parallelogram lateral faces. Figure (b) is a pyramid, which has only one base and triangular lateral faces.

Therefore, Figure (b) is not a prism.

The correct option is (b).

Explanation:

A pyramid is a polyhedron formed by connecting a polygonal base to an apex. The lateral faces of a pyramid are triangles, and the number of lateral faces is equal to the number of sides of the base polygon.

We are given 4 congruent equilateral triangles. These triangles will form the lateral faces of the pyramid.

Since there are 4 lateral faces, the base of the pyramid must be a polygon with 4 sides.

A polygon with 4 sides is a quadrilateral.

Since the 4 triangular faces are congruent and equilateral, the length of their bases (which form the edges of the base polygon) must all be equal. Therefore, the base polygon must be a quadrilateral with all sides equal.

A quadrilateral with all sides equal is either a rhombus or a square.

To form a standard pyramid where the apex is directly above the center of the base (a right pyramid), and given congruent lateral faces, the base is typically a regular polygon. The regular quadrilateral is a square.

For the triangular faces to fit perfectly onto the base, the side length of the square base must be the same as the side length of the equilateral triangles.

Let's examine the options:

(a) An equilateral triangle: A pyramid with a triangular base (a tetrahedron) has 4 faces, all of which are triangles. If the 4 given triangles already form the faces of a tetrahedron, nothing more is needed. However, the question asks what is needed *more*, implying the 4 triangles are just the lateral faces and the base is missing. If the base is a triangle, there would only be 3 lateral faces.

(b) A square with same side length as of triangle: This would provide a 4-sided base. If the side length of the square is the same as the side length of the equilateral triangles, the triangles can be attached to each side of the square to form a square pyramid. This fits the criteria.

(c) 2 equilateral triangles with side length same as triangle: This would result in a total of 6 triangles. This does not form a standard pyramid structure with 4 lateral faces.

(d) 2 squares with side length same as triangle: A pyramid has only one base. Adding two squares would not form a pyramid.

Therefore, to make a pyramid with 4 congruent equilateral triangles as lateral faces, a square with the same side length as the triangle is needed for the base.

The correct option is (b) A square with same side length as of triangle.

Explanation:

We are given the actual side length of a square garden and the scale used to draw its picture. We need to find the perimeter of the square in the picture.

Given:

Actual side length of square garden = 30 m

Scale used for the picture = 1 cm : 5 m

To Find:

The perimeter of the square in the picture.

Solution:

First, we need to determine the side length of the square in the picture using the given scale.

The scale 1 cm : 5 m means that every 5 meters of actual length are represented by 1 centimeter in the picture.

To find the side length in centimeters in the picture, we use the proportion from the scale:

Rearranging to find the side length in the picture:

Substitute the given actual side length (30 m):

Now, we can calculate the perimeter of the square in the picture.

The formula for the perimeter of a square is $P = 4 \times s$, where $s$ is the side length.

Substitute the side length from (i):

The perimeter of the square garden in the picture is 24 cm.

The correct option is (b) 24 cm.

Explanation:

A vertex (plural: vertices) is a corner point of a geometric shape. In 3D shapes, it is where edges meet or where a curved surface tapers to a point.

Let's examine each figure:

Figure (a): This figure is a cylinder. A cylinder has two circular bases and a curved lateral surface. It has edges (where the bases meet the curved surface) but no vertices (points where edges meet or where a curved surface tapers to a point).

Figure (b): This figure is a sphere. A sphere is a perfectly round 3D shape with a continuous curved surface. It has no edges or vertices.

Figure (c): This figure is a cone. A cone has a circular base and a curved lateral surface that tapers to a single point called the apex. The apex is a vertex of the cone.

Figure (d): This figure is a torus (like a donut). It is a shape with a continuous curved surface. It has no edges or vertices.

Among the given options, only the cone has a vertex (the apex).

The correct option is (c).

Explanation:

We are given the scale used in a map and the distance between the school and the book shop on the map. We need to find the actual distance between these two places.

Given:

Scale of the map = 1 cm : 0.5 km

Distance between School and Book Shop on the map = 2.5 cm (from the provided image)

To Find:

The actual distance (in km) between School and the Book Shop.

Solution:

The scale 1 cm : 0.5 km means that every 1 centimeter on the map represents an actual distance of 0.5 kilometers on the ground.

To find the actual distance, we can set up a ratio based on the scale:

We are given the distance on the map (2.5 cm), so we can substitute this value into the equation to find the actual distance:

Now, we can solve for the Actual Distance:

Performing the multiplication:

The actual distance between the School and the Book Shop is 1.25 km.

The correct option is (a) 1.25.

Explanation:

For any convex polyhedron (and many non-convex polyhedra), Euler's formula must hold true. Euler's formula is:

where $F$ is the number of faces, $V$ is the number of vertices, and $E$ is the number of edges.

We need to check which of the given options does not satisfy Euler's formula.

(a) V = 4, F = 4, E = 6

Substitute into Euler's formula:

Since $2 = 2$, this set of values satisfies Euler's formula. This corresponds to a tetrahedron (a triangular pyramid).

(b) V = 6, F = 8, E = 12

Substitute into Euler's formula:

Since $2 = 2$, this set of values satisfies Euler's formula. This corresponds to an octahedron.

(c) V = 20, F = 12, E = 30

Substitute into Euler's formula:

Since $2 = 2$, this set of values satisfies Euler's formula. This corresponds to a dodecahedron.

(d) V = 4, F = 6, E = 6

Substitute into Euler's formula:

Since $4 \neq 2$, this set of values does not satisfy Euler's formula. Therefore, a polyhedron cannot have these numbers of faces, vertices, and edges.

The correct option is (d) V = 4, F = 6, E = 6.

Explanation:

We are given the height of a room in a blueprint and the actual height of the room. We need to determine the scale used in the blueprint.

Given:

Height of the room in the blueprint = 33 cm

Actual height of the room = 330 cm

To Find:

The scale used, in the format 1 : x.

Solution:

The scale of a drawing or blueprint represents the ratio of a length on the drawing to the corresponding actual length. It can be expressed as:

Substitute the given values for the height of the room:

To express the scale in the format 1 : x, we simplify the fraction by dividing both the numerator and the denominator by the numerator (33):

This ratio can be written as a scale of 1 : 10.

This means that 1 cm on the blueprint represents an actual distance of 10 cm in the room.

The scale used by the architect is 1 : 10.

The correct option is (b) 1 : 10.

Explanation:

We need to find the number of hospitals in the town based on the provided map.

To do this, we should look at the map and its legend to identify how hospitals are represented.

In the legend shown on the map, there is an entry for "HOSPITAL". The symbol used for the hospital is a red cross within a square.

Now, we need to scan the map to find instances of this symbol or the label "HOSPITAL".

Upon examining the map, we can locate the label "HOSPITAL" in one place.

Counting the number of times "HOSPITAL" is marked on the map, we find there is only 1 hospital.

Therefore, the number of hospitals in the town is 1.

The correct option is (a) 1.

Explanation:

We need to find the ratio of the number of general stores to the number of grounds in the town, based on the provided map.

First, let's count the number of general stores in the town. Look at the map and the legend for the symbol or label representing a "General Store".

In the legend, "General Store" is represented by a symbol. Let's locate this symbol or the label "General Store" on the map.

Scanning the map, we can find the label "General Store" in two places.

Number of General Stores = 2

Next, let's count the number of grounds in the town. Look at the map and the legend for the symbol or label representing a "Ground".

In the legend, "Ground" is represented by the label "GROUND". Let's locate this label on the map.

Scanning the map, we can find the label "GROUND" in one place.

Number of Grounds = 1

Now, we need to find the ratio of the number of general stores and that of the ground. The ratio is expressed as (Number of General Stores) : (Number of Grounds).

The ratio of the number of general stores and that of the ground is 2 : 1.

The correct option is (b) 2 : 1.

Explanation:

We need to find the number of schools in the town based on the provided map.

To do this, we should look at the map and its legend to identify how schools are represented.

In the legend shown on the map (referencing the image from Question 19 which is used for questions 19-21), there is an entry for "SCHOOL". The symbol used for the school is a building with a flag on top.

Now, we need to scan the map to find instances of this symbol or the label "SCHOOL".

Upon examining the map carefully, we can locate the label and symbol for "SCHOOL" in the following places:

• Near the top-left area of the map.
• In the middle-left area of the map.
• In the bottom-right area of the map.

Counting the number of times "SCHOOL" is marked on the map, we find there are 3 schools.

Therefore, according to the map, the number of schools in the town is 3.

The correct option is (b) 3.

Explanation:

A square prism is a prism with a square base. Its faces are two congruent square bases and four rectangular lateral faces. If the rectangular lateral faces are also squares (meaning the height of the prism is equal to the side length of the square base), then all six faces are squares.

A solid shape with six congruent square faces is known as a cube.

Therefore, a square prism where the height equals the side of the square base is also called a cube.

The blank should be filled with the word "cube".

Answer: Square prism is also called a cube.

Explanation:

A rectangular prism is a prism that has a rectangular base. It is a three-dimensional solid with six faces, where all the faces are rectangles. The opposite faces are congruent and parallel.

A solid shape bounded by six rectangular faces is also known as a cuboid.

Therefore, a rectangular prism is also called a cuboid.

The blank should be filled with the word "cuboid".

Answer: Rectangular prism is also called a cuboid.

Explanation:

The provided image shows a three-dimensional solid, which appears to be a triangular pyramid (tetrahedron).

We need to find the number of faces meeting at the vertex labeled B.

Let's analyze the structure of the triangular pyramid from the image.

The vertices are labeled A, B, C, and D.

The faces are the triangles that form the surface of the solid: $\triangle ABC$, $\triangle ABD$, $\triangle ACD$, and $\triangle BCD$.

We are interested in the vertex B.

Let's identify which faces share the vertex B:

• Face 1: $\triangle ABC$ (This triangle has vertices A, B, and C. So, B is a vertex of this face).
• Face 2: $\triangle ABD$ (This triangle has vertices A, B, and D. So, B is a vertex of this face).
• Face 3: $\triangle BCD$ (This triangle has vertices B, C, and D. So, B is a vertex of this face).
• Face 4: $\triangle ACD$ (This triangle has vertices A, C, and D. It does not have B as a vertex).

Thus, the faces meeting at vertex B are $\triangle ABC$, $\triangle ABD$, and $\triangle BCD$.

Counting these faces, we find that 3 faces meet at vertex B.

The blank should be filled with the number 3.

Answer: In the figure, the number of faces meeting at B is 3.

Explanation:

A pyramid is a polyhedron formed by connecting a polygonal base to a single point called the apex.

The faces of a pyramid consist of two types:

1. The base: This is the polygon at the bottom. If the base is an n-sided polygon, there is 1 base face.

2. The lateral faces: These are the triangular faces that connect the sides of the base to the apex. Since there are n sides in the base polygon, there will be n triangular lateral faces.

To find the total number of faces, we add the number of base faces and the number of lateral faces:

So, a pyramid on an n-sided polygon has $n+1$ faces.

The blank should be filled with the expression $(n+1)$ or "n+1".

Answer: A pyramid on an n sided polygon has n+1 faces.

Explanation:

We are given the number of faces ($F$) and the number of vertices ($V$) of a solid shape. We need to find the number of edges ($E$).

Given:

Number of faces, F = 12

Number of vertices, V = 20

To Find:

The number of edges, E.

Solution:

Assuming the solid shape is a polyhedron (for which the question is applicable), we can use Euler's formula:

Substitute the given values of $F$ and $V$ into Euler's formula:

Simplify the equation:

To find $E$, rearrange the equation:

So, the number of edges in this solid is 30.

This solid is an icosahedron, a regular polyhedron with 20 triangular faces, 12 vertices, and 30 edges.

The blank should be filled with the number 30.

Answer: If a solid shape has 12 faces and 20 vertices, then the number of edges in this solid is 30.

Explanation:

The given net consists of a rectangle and two congruent circles attached to opposite sides of the rectangle.

Let's consider how this net folds into a 3D shape:

• The rectangle will form the curved lateral surface of the solid. When the rectangle is rolled up, its width becomes the height of the solid, and its length becomes the circumference of the bases.
• The two circles, attached to the sides corresponding to the length of the rectangle, will form the top and bottom bases of the solid. Since they are circles, the bases will be circular.

A three-dimensional solid with two parallel and congruent circular bases connected by a curved lateral surface is called a cylinder.

Therefore, the given net can be folded to make a cylinder.

The blank should be filled with the word "cylinder".

Answer: The given net can be folded to make a cylinder.

Explanation:

We are looking for a solid figure that has exactly one vertex.

Let's consider some common solid shapes and their number of vertices:

• Cube: 8 vertices
• Cuboid: 8 vertices
• Triangular Prism: 6 vertices
• Square Pyramid: 5 vertices (4 at the base, 1 at the apex)
• Triangular Pyramid (Tetrahedron): 4 vertices
• Cylinder: 0 vertices (edges are circles, no sharp corners)
• Sphere: 0 vertices (continuous curved surface)
• Cone: 1 vertex (the apex where the curved surface meets)

From this list, the cone is a solid figure that has only one vertex.

A cone tapers from a base to a single point (the apex), which is its only vertex.

The blank should be filled with the word "cone".

Answer: A solid figure with only 1 vertex is a cone.

Explanation:

We are given that a pyramid has eight edges. We need to find the total number of faces in this pyramid.

Let the base of the pyramid be a polygon with $n$ sides.

In a pyramid, the edges consist of the edges of the base polygon and the lateral edges connecting each vertex of the base to the apex.

• Number of edges in the base = $n$
• Number of lateral edges = $n$ (one for each vertex of the base)

The total number of edges ($E$) in a pyramid is the sum of the base edges and the lateral edges:

We are given that the number of edges is 8. So, we have:

Solving for $n$:

So, the base of the pyramid is a 4-sided polygon (a quadrilateral, likely a square for a standard pyramid).

Now, let's find the number of faces ($F$) in this pyramid.

The faces of a pyramid consist of the base face and the lateral triangular faces.

• Number of base faces = 1 (the n-sided polygon)
• Number of lateral faces = n (one triangular face for each side of the base)

The total number of faces is the sum of the base face and the lateral faces:

Substitute the value of $n=4$ from (iii) into (iv):

So, a pyramid with eight edges has 5 faces.

This corresponds to a square pyramid, which has 1 square base and 4 triangular faces ($1+4=5$ faces), 4 base vertices + 1 apex vertex = 5 vertices, and 4 base edges + 4 lateral edges = 8 edges. Let's check Euler's formula for this: F + V - E = 5 + 5 - 8 = 10 - 8 = 2. The formula holds.

The blank should be filled with the number 5.

Answer: Total number of faces in a pyramid which has eight edges is 5.

Explanation:

A rectangular prism is a polyhedron with six faces. All six faces of a rectangular prism are rectangles. Some or all of these rectangles might also be squares, but since a square is a specific type of rectangle, we can say that all faces are rectangles.

Let's count the number of faces in a rectangular prism.

A rectangular prism has three pairs of congruent and parallel rectangular faces:

• The top and bottom bases (rectangles).
• The front and back faces (rectangles).
• The left and right faces (rectangles).

In total, there are $2 + 2 + 2 = 6$ faces.

Since every face of a rectangular prism is a rectangle, the net of a rectangular prism will consist of the shapes of these six faces laid out in a connected pattern in a plane.

Therefore, the net of a rectangular prism has 6 rectangles.

The hint reminds us that if some of the faces are squares, they are still considered rectangles for the purpose of counting the total number of rectangular faces.

The blank should be filled with the number 6.

Answer: The net of a rectangular prism has 6 rectangles.

Explanation:

In geometry, a diagonal connects two non-adjacent vertices. In a three-dimensional shape (a solid), there are two types of diagonals:

1. Face diagonal: A line segment connecting two vertices that lie on the same face but are not adjacent (i.e., they are opposite vertices of that face). These diagonals lie within a single face.

2. Space diagonal (or interior diagonal): A line segment connecting two vertices that do not lie on the same face. These diagonals pass through the interior of the solid.

The statement "In a three-dimensional shape, diagonal is a line segment that joins two vertices that do not lie on the ______ face" describes a space diagonal.

The vertices joined by a space diagonal belong to different faces, specifically faces that are not adjacent or share a common edge.

The vertices do not lie on the **same** face.

The blank should be filled with the word "same".

Answer: In a three-dimensional shape, diagonal is a line segment that joins two vertices that do not lie on the same face.

Explanation:

We are given a scale where 4 km on a map is represented by 1 cm. We need to find out how many centimeters represent 16 km on this map.

Given:

Scale: 4 km on map = 1 cm

To Find:

The representation in cm for 16 km.

Solution:

The scale can be expressed as a ratio:

We want to find the distance on the map for an actual distance of 16 km. Let the distance on the map be $x$ cm.

To solve for $x$, we can cross-multiply or multiply both sides by 16 km:

So, 16 km is represented by 4 cm on the map.

The blank should be filled with the number 4.

Answer: If 4 km on a map is represented by 1 cm, then 16 km is represented by 4 cm.

Explanation:

We are given the actual distance between two places and the distance representing it on a map. We need to find the scale used for the map.

Given:

Actual distance = 110 km

Distance on map = 25 mm

To Find:

The scale used, typically expressed in the format 1 : x (unitless ratio) or 1 unit : y units.

Solution:

The scale is the ratio of the distance on the map to the actual distance.

We need to express both distances in the same unit. Let's convert kilometers to millimeters.

We know that:

Combining these conversions:

Now, convert the actual distance from km to mm:

Now, calculate the ratio of map distance to actual distance:

Simplify the fraction to express the scale in the format 1 : x. Divide both numerator and denominator by 25:

Let's perform the division $110,000,000 \div 25$:

$110,000,000 \div 25 = (1100 \times 100000) \div 25 = (1100 \div 25) \times 100000$

$1100 \div 25 = 44$

So, $110,000,000 \div 25 = 44 \times 100000 = 4,400,000$

The scale is 1 : 4,400,000.

The blank should be filled with the scale 1 : 4,400,000.

Answer: If actual distance between two places A and B is 110 km and it is represented on a map by 25 mm. Then the scale used is 1 : 4,400,000.

Explanation:

A pentagonal prism is a prism with a pentagonal base.

The faces of a prism consist of two types:

1. The bases: There are two congruent and parallel polygonal bases. Since the base is a pentagon, there are 2 pentagonal faces.

2. The lateral faces: These are the faces that connect the corresponding sides of the two bases. In a prism, the lateral faces are parallelograms (usually rectangles in a right prism). The number of lateral faces is equal to the number of sides of the base polygon. Since the base is a pentagon (a 5-sided polygon), there are 5 lateral faces.

To find the total number of faces ($F$) in a pentagonal prism, we add the number of base faces and the number of lateral faces:

So, a pentagonal prism has 7 faces.

Let's verify using Euler's formula: F + V - E = 2.

Vertices ($V$): A pentagon has 5 vertices. Since there are two bases, the total number of vertices is $5 \times 2 = 10$.

Edges ($E$): A pentagonal base has 5 edges. There are 2 bases, contributing $5 \times 2 = 10$ edges. There are also 5 lateral edges connecting the corresponding vertices of the bases. Total edges = $10 + 5 = 15$.

Check Euler's formula: F + V - E = 7 + 10 - 15 = 17 - 15 = 2. The formula holds, confirming our face count.

The blank should be filled with the number 7.

Answer: A pentagonal prism has 7 faces.

Explanation:

We are considering a pyramid with a hexagonal base. We need to find the number of vertices in this pyramid.

The vertices of a pyramid consist of two types:

1. The vertices of the base polygon. Since the base is a hexagon, and a hexagon is a 6-sided polygon, it has 6 vertices.

2. The apex: This is the single vertex at the top where all the lateral faces meet.

To find the total number of vertices ($V$) in a hexagonal pyramid, we add the number of vertices in the base and the apex vertex:

So, a pyramid with a hexagonal base has 7 vertices.

The blank should be filled with the number 7.

Answer: If a pyramid has a hexagonal base, then the number of vertices is 7.

The first image shows a rectangle.

The second image shows a 3D object which looks like a house with a rectangular base and a triangular prism on top.

When looking at the second object from the top, we only see the rectangular base.

Therefore, the first image is the top view of the second image.

The blank should be filled with top.

Let's count the number of cubes layer by layer, starting from the bottom.

The bottom layer appears to be a $2 \times 2$ square arrangement of cubes.

Number of cubes in the bottom layer = $2 \times 2 = 4$.

The second layer from the bottom appears to have 2 cubes arranged in a $2 \times 1$ row on top of part of the bottom layer.

Number of cubes in the second layer = $2 \times 1 = 2$.

The top layer appears to have 1 cube on top of part of the second layer.

Number of cubes in the top layer = $1$.

Total number of cubes = (Cubes in bottom layer) + (Cubes in second layer) + (Cubes in top layer)

Total number of cubes = $4 + 2 + 1 = 7$.

The blank should be filled with 7.

Let V be the number of vertices, F be the number of faces, and E be the number of edges in a polyhedron.

We are given that the sum of the number of vertices and faces is 14.

According to Euler's formula for convex polyhedrons, the relationship between the number of vertices, faces, and edges is:

We can rewrite Euler's formula as:

(V + F) - E = 2

Now, substitute the value of (V + F) from equation (i) into this rearranged formula:

$14 - \text{E} = 2$

To find the number of edges E, rearrange the equation:

$\text{E} = 14 - 2$

$\text{E} = 12$

So, the number of edges in the polyhedron is 12.

The blank should be filled with 12.

A regular polyhedron is a convex polyhedron where all faces are congruent regular polygons and the same number of faces meet at each vertex.

These are also known as Platonic solids.

There are exactly five convex regular polyhedra:

• The tetrahedron (4 faces, equilateral triangles)
• The cube (6 faces, squares)
• The octahedron (8 faces, equilateral triangles)
• The dodecahedron (12 faces, regular pentagons)
• The icosahedron (20 faces, equilateral triangles)

Thus, the total number of regular polyhedra is 5.

The blank should be filled with 5.

A regular polyhedron is a convex polyhedron that is highly symmetrical. By definition, all its faces are identical (congruent) and are regular polygons.

A regular polygon is a polygon that is both equiangular (all angles are equal) and equilateral (all sides are equal).

For example, the faces of a cube are congruent squares (regular polygons), and the faces of a regular tetrahedron are congruent equilateral triangles (regular polygons).

Therefore, a regular polyhedron is a solid made up of faces that are congruent regular polygons.

The blank should be filled with congruent regular polygonal.

For solid (a) (image 23.png):

The views are shown in image 24.png (Left, Middle, Right).

Front View: The right image in 24.png (Stack of 3 rectangles).

Side View: The middle image in 24.png (L-shape).

Top View: The left image in 24.png (Rectangle).

For solid (b) (image 25.png):

The views are shown in image 26.png (Left, Middle, Right).

Front View: The middle image in 26.png (Stepped shape).

Side View: The left image in 26.png (Rectangle).

Top View: The right image in 26.png (Stepped shape).

For solid (c) (image 27.png):

The views are shown in image 28.png (Left, Middle, Right).

Front View: The middle image in 28.png (Two side-by-side rectangles of different heights).

Side View: The right image in 28.png (Single rectangle).

Top View: The left image in 28.png (Single rectangle).

For solid (d) (image 29.png):

The views are shown in image 30.png (Left, Middle, Right).

Front View: The middle image in 30.png (Stack of 2 rectangles).

Side View: The right image in 30.png (Single rectangle).

Top View: The left image in 30.png (L-shape).

False

A cuboid is a three-dimensional solid with six rectangular faces. It is also known as a rectangular prism.

A tetrahedron is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. It is the simplest of all the convex polyhedra.

Since a cuboid has 6 faces and a tetrahedron has 4 faces, they are different shapes and one is not another name for the other.

False

A polyhedron is a three-dimensional solid whose surface is made up of a finite number of polygonal faces. The smallest possible number of faces a polyhedron can have is 4.

The simplest polyhedron is the tetrahedron, which has 4 triangular faces, 6 edges, and 4 vertices.

According to Euler's formula for convex polyhedra, the number of vertices ($V$), edges ($E$), and faces ($F$) are related by the equation:

Each edge of a polyhedron is shared by exactly two faces. This means that twice the number of edges is equal to the sum of the number of edges of all faces. Since the minimum number of edges for any face is 3 (as faces are polygons), we have:

If a polyhedron had 3 faces ($F=3$), then from the inequality $2E \geq 3F$, we would get $2E \geq 3(3) = 9$, which means $E \geq 4.5$. Since the number of edges must be an integer, the minimum number of edges would be $E = 5$.

Now, substituting $F=3$ and $E=5$ into Euler's formula $V - E + F = 2$:

So, if a polyhedron had 3 faces, it would require 5 edges and 4 vertices. However, it is not possible to form a closed three-dimensional figure (a polyhedron) with only 4 vertices and 5 edges where each face is a polygon and each edge is shared by exactly two faces. The minimum number of edges connecting 4 vertices to form a polyhedron is 6 (as in a tetrahedron).

Therefore, a polyhedron must have at least 4 faces.

True

As discussed in the previous question, the minimum number of faces a polyhedron can have is 4.

A polyhedron with 4 faces is called a tetrahedron.

A triangular pyramid is a pyramid with a triangle as its base. It has 1 triangular base and 3 triangular lateral faces. Therefore, a triangular pyramid has a total of $1 + 3 = 4$ faces.

A tetrahedron is the same geometric shape as a triangular pyramid. Both are polyhedra with the minimum possible number of faces (4).

True

A regular octahedron is one of the five Platonic solids. It is a polyhedron with 8 faces, 12 edges, and 6 vertices.

The faces of a regular octahedron are 8 congruent equilateral triangles.

An isosceles triangle is a triangle that has at least two sides of equal length. An equilateral triangle is a triangle in which all three sides are equal in length. Since an equilateral triangle has all three sides equal, it also has at least two sides equal, meaning every equilateral triangle is also an isosceles triangle.

Therefore, since the faces of a regular octahedron are congruent equilateral triangles, they are also congruent isosceles triangles. The statement is true.

False

A prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy of the first, and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases.

A pentagonal prism is a prism with a pentagonal base. This means it has two congruent and parallel pentagonal faces (the bases).

The faces connecting the two pentagonal bases are called lateral faces. Since a pentagon has 5 sides, there are 5 lateral faces, which are rectangles (or parallelograms in a non-right prism).

So, a pentagonal prism has:

• 2 pentagonal faces (the bases)
• 5 rectangular faces (the lateral faces)

The total number of faces is $2 + 5 = 7$.

The statement claims that a pentagonal prism has 5 pentagons, which is incorrect. It has only 2 pentagons.

False

A cylinder is a three-dimensional solid with two congruent and parallel circular bases and a curved lateral surface.

A prism is a polyhedron whose bases are two parallel and congruent polygons and whose lateral faces are parallelograms.

While a cylinder shares the property of having two opposite congruent and parallel bases with a prism, the key difference lies in the lateral surface. A prism's lateral surface is composed of flat polygonal faces, whereas a cylinder's lateral surface is curved.

A prism is a type of polyhedron, which means all its faces must be polygons (flat surfaces with straight edges). A cylinder is not a polyhedron because it has a curved surface.

Therefore, despite having congruent circular bases, a cylinder is not classified as a prism.

False

Euler's formula relates the number of vertices ($V$), edges ($E$), and faces ($F$) of certain three-dimensional shapes. The formula is:

This formula is specifically true for convex polyhedra (three-dimensional solids whose faces are polygons) and some non-convex polyhedra that are topologically equivalent to a sphere.

However, there are many three-dimensional shapes for which this formula does not apply, such as:

• Shapes with curved surfaces (like spheres, cylinders, cones). These shapes do not have distinct vertices, edges, and faces in the way polyhedra do.
• Polyhedra with holes (like a torus or a hollow cube). For these shapes, the formula becomes $V - E + F = 2 - 2g$, where $g$ is the number of "holes" (genus).

Since Euler's formula ($V - E + F = 2$) is not applicable to all three-dimensional shapes (e.g., shapes with curved surfaces), the statement that it is true for all three-dimensional shapes is false.

False

For any simple polyhedron (one without holes), the relationship between the number of vertices ($V$), edges ($E$), and faces ($F$) is given by Euler's formula:

In the given statement, we have:

$F = 10$ (Number of faces)

$E = 20$ (Number of edges)

$V = 15$ (Number of vertices)

Let's substitute these values into Euler's formula:

According to Euler's formula for simple polyhedra, the result must be 2 ($V - E + F = 2$). In this case, we got 5, which is not equal to 2.

This set of values for $V$, $E$, and $F$ does not satisfy Euler's formula for a simple polyhedron, nor for a polyhedron with a non-negative integer number of holes ($V - E + F = 2 - 2g$, where $g$ is the number of holes, requires $5 = 2 - 2g$, which gives $g = -3/2$, not a valid number of holes).

Therefore, a polyhedron cannot have 10 faces, 20 edges, and 15 vertices.

True

The top view of an object is what you see when you look at it from directly above.

The first image shows a three-dimensional structure likely made of unit cubes. It appears to have a base layer that forms a rectangle, and additional cubes stacked on top.

Assuming the base layer is a $3 \times 2$ arrangement of cubes and the additional cubes are placed on top within the boundaries of this base, when viewed from directly above, the shape visible will be the outline of the base layer.

The visible area from the top will be a solid rectangle corresponding to the dimensions of the base layer.

The second image is a simple rectangle. If the base of the structure in the first image is rectangular, then its top view will be a rectangle.

Based on the typical representation of such structures with unit cubes, the base is rectangular (likely $3 \times 2$). The top view of such a structure is indeed a rectangle of the same dimensions as the base.

False

A parallelogram is a plane figure (a two-dimensional shape) with four straight sides where opposite sides are parallel and equal in length.

In geometry, the term edges typically refers to the line segments where two faces of a three-dimensional object meet.

Two-dimensional shapes like parallelograms have sides, not edges. A parallelogram has 4 sides.

While the boundaries of a 2D shape can sometimes be referred to loosely as edges in certain contexts (like computer graphics), in formal geometric terms, "edge" is a property of 3D objects (polyhedra, etc.). Since a parallelogram is a 2D shape, it does not have edges in the same way a cube or a prism does.

Therefore, the statement is false because a parallelogram has 4 sides, not 4 edges (in the context of 3D geometry).

False

A net is a two-dimensional pattern that can be folded to form a three-dimensional solid shape.

Many solid shapes can be formed by folding different net patterns. This means that a single solid shape often has multiple possible nets, not just one unique net.

For example, a cube (a common solid shape) can be unfolded into several different 2D patterns. There are exactly 11 distinct nets that can be folded to form a cube.

Since solid shapes like the cube have more than one net, the statement that every solid shape has a unique net is false.

False

In geometry, a diagonal is a line segment joining two non-adjacent vertices of a polygon or a polyhedron.

For a pyramid, we consider diagonals on its faces and potentially space diagonals.

The faces of a pyramid consist of a base polygon and triangular lateral faces. The triangular lateral faces do not have diagonals, as a triangle has no non-adjacent vertices.

The base of a pyramid is a polygon. If the base polygon has 4 or more sides (e.g., a square, a pentagon, a hexagon, etc.), it will have diagonals. For example, a square base has two diagonals connecting opposite corners on the base.

Pyramids generally do not have space diagonals (connecting two vertices that do not lie on the same face), as any line segment connecting the apex to a base vertex is an edge, and any line segment connecting two base vertices is either a base edge or a base diagonal.

Since a pyramid with a base of 4 or more sides has diagonals on its base face, the statement "Pyramids do not have a diagonal" is false.

I am unable to provide a definitive answer to this question.

The question refers to a "given shape", but the image or description of this shape is not included in the input provided.

To determine if the statement "The given shape is a cylinder" is true or false, I need to see the specific shape being referred to.

A cylinder is a three-dimensional solid with two congruent and parallel circular bases and a curved lateral surface.

Please provide the image or a description of the shape so I can answer the question correctly.

True

A cuboid is a three-dimensional solid with six rectangular faces.

In a cuboid, we typically consider two types of diagonals:

• Face diagonals: These lie on the faces and connect two non-adjacent vertices on the same face.
• Space diagonals: These pass through the interior of the cuboid and connect two vertices that do not lie on the same face.

Let's consider the space diagonals of a cuboid. A cuboid has 8 vertices. A space diagonal connects a vertex to the opposite vertex that is furthest away and not on the same face.

From any one vertex, there is exactly one other vertex that is diagonally opposite through the interior of the cuboid.

Since there are 8 vertices, and each space diagonal connects two vertices, the total number of space diagonals is $\frac{8}{2} = 4$.

Thus, a cuboid has exactly 4 space diagonals. Since 4 is "at least 4", the statement is true.

(For completeness, a cuboid also has 12 face diagonals, 2 on each of the 6 faces. So, in total, a cuboid has $4 + 12 = 16$ diagonals if both types are counted. In either case, it has at least 4 diagonals.)

True

A cube is a three-dimensional solid bounded by six congruent square faces, with three meeting at each vertex.

A prism is a polyhedron comprising an n-sided polygonal base, a second base which is a translated copy of the first, and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases.

Let's examine if a cube fits the definition of a prism:

• A cube has two opposite faces that are parallel and congruent squares. These can be considered the bases of the prism. A square is a polygon with $n=4$ sides.
• The faces connecting these two square bases are also squares. A square is a type of rectangle, and a rectangle is a type of parallelogram. There are 4 such lateral faces, corresponding to the 4 sides of the square base.

Since a cube has two parallel and congruent polygonal bases (squares) and lateral faces that are parallelograms (squares/rectangles), it fits the definition of a prism. Specifically, a cube is a type of square prism where the height is equal to the side length of the square base.

Therefore, the statement "All cubes are prisms" is true.

False

A cylinder is a three-dimensional geometric shape that has two parallel and congruent (identical in size and shape) circular bases connected by a curved lateral surface.

The key property of a cylinder's bases is that they are congruent circles. Congruent circles have the same radius.

The statement claims that a cylinder has two circular faces of different radii, which contradicts the definition of a cylinder.

A 3-D shape with two circular faces of different radii and a curved surface connecting them would typically be a frustum of a cone or a truncated cone, not a standard cylinder.

True

The figure shows the net of a cylinder. The net typically consists of two congruent circles (which form the top and bottom bases of the cylinder) and one rectangle (which forms the curved lateral surface when rolled up).

When the rectangular part of the net is rolled up to form the curved surface of the cylinder, the two opposite edges of the rectangle meet and join the circumferences of the two circular bases.

This means that the length of the rectangle must be exactly equal to the distance around the circular base, which is the circumference of the circle.

If the radius of the circular base is $r$, the circumference of the circle is $2\pi r$. The length of the rectangle in the net is equal to this circumference, i.e., Length $= 2\pi r$. The width of the rectangle is the height of the cylinder, say $h$.

Therefore, the statement that the length of the rectangle in the net of a cylinder is the same as the circumference of the circles in its net is correct.

True

The given scale on the map is that 1 cm represents an actual distance of 100 m.

This can be written as a ratio:

To find the actual distance corresponding to 2 cm on the map, we can multiply both sides of the ratio by 2:

So, 2 cm on the map represents an actual distance of 200 m.

The statement is consistent with this calculation.

True

The problem provides the height of the model ship and the scale used to create the model.

Given information:

• Height of the model = 3.5 cm
• Scale = 1:60

The scale 1:60 means that every unit of measurement on the model represents 60 units of the same measurement in reality. In this case, 1 cm on the model represents 60 cm of the actual ship's size.

To find the actual height of the ship, we multiply the model height by the scale factor:

Calculating the product:

So, the actual height of the ship is 210 cm.

The calculated actual height (210 cm) matches the height given in the statement (210 cm).

Therefore, the statement is true.

True

The problem involves interpreting a map or drawing scale to find an actual dimension.

Given information:

• Actual width of the store room = 280 cm
• Scale of the drawing = 1:7

A scale of 1:7 means that every 1 unit of measurement on the drawing represents 7 units of the same measurement in reality.

Let $W_{drawing}$ be the width of the room in the drawing and $W_{actual}$ be the actual width of the room.

The scale can be written as a ratio:

We are given $W_{actual} = 280 \text{ cm}$. We need to find $W_{drawing}$.

Rearranging the formula:

The calculated width of the room in the drawing is 40 cm, which matches the value given in the statement.

Therefore, the statement is true.