Solid Shapes (3D Geometry): Introduction and Types

2-Dimensional vs 3-Dimensional Shapes: Distinction

In the world of geometry, shapes are fundamentally distinguished by the number of dimensions required to describe them fully. This leads to the classification of shapes as either two-dimensional (2D) or three-dimensional (3D).

2-Dimensional (2D) Shapes

2-Dimensional shapes, often referred to as plane figures or flat shapes, exist entirely within a single plane. They possess only two primary dimensions:

• Length
• Width (or Breadth)

These shapes have no thickness or depth. Imagine drawing a shape on a piece of paper; that shape is 2-dimensional. While it might be on a physical paper (which has thickness), the geometric shape itself is considered to be perfectly flat.

For 2D shapes, we typically measure:

• The length of their boundary, which is called the perimeter.
• The amount of flat space they cover, which is called the area.

Common examples of 2D shapes include:

• Triangles (Equilateral, Isosceles, Scalene, Right, etc.)
• Quadrilaterals (Square, Rectangle, Parallelogram, Rhombus, Trapezium, Kite)
• Polygons with more sides (Pentagon, Hexagon, Octagon, etc.)
• Circle

3-Dimensional (3D) Shapes

3-Dimensional shapes, also known as solid shapes or simply solids, exist in three-dimensional space. Unlike 2D shapes, they have depth and occupy a volume. They require three primary dimensions for their description:

• Length
• Width (or Breadth)
• Height (or Depth)

These shapes have thickness and can be physically held (though we represent them visually in 2D). They fill up a portion of space.

For 3D shapes, we typically measure:

• The total area of their outer surfaces, called the surface area.
• The amount of space they occupy, called the volume.

Common examples of 3D shapes include:

• Cube
• Cuboid
• Cylinder
• Cone
• Sphere
• Pyramid
• Prism

Key Differences Summarized

The fundamental differences between 2D and 3D shapes can be clearly seen when comparing their characteristics:

Understanding this distinction is crucial as it forms the basis for studying various geometric concepts, from area and perimeter in 2D to surface area and volume in 3D, as well as visualising objects in space.

Introduction to Solid Shapes (Cubes, Cuboids, Cylinders, Cones, Spheres, Pyramids)

Moving from two-dimensional plane figures, we now introduce the concept of solid shapes, which are three-dimensional objects that occupy space. These shapes are part of our everyday experience and form the basis of stereometry, the study of three-dimensional geometry. Here, we will explore some of the most fundamental and common types of solid shapes.

1. Cube

A cube is one of the simplest and most regular polyhedra. It is a solid object bounded by six square faces.

• It has 6 faces, and all are identical squares.
• It has 12 edges, and all are of equal length.
• It has 8 vertices, where three edges meet.

The angle between any two adjacent faces of a cube is $90^\circ$. A cube is a special case of a cuboid and is also known as a regular hexahedron (a polyhedron with 6 faces). Think of a standard dice; that's a perfect example of a cube.

2. Cuboid (or Rectangular Prism)

A cuboid is a solid shape with six rectangular faces. It is also known as a rectangular prism.

• It has 6 faces, and opposite faces are identical rectangles.
• It has 12 edges. Edges come in sets of four equal lengths (corresponding to the length, width, and height of the cuboid).
• It has 8 vertices.

Unlike a cube, the edges meeting at a vertex can have different lengths. A cuboid can be thought of as a stretched or flattened cube. Examples include bricks, matchboxes, and most rectangular boxes.

3. Cylinder (Right Circular Cylinder)

A cylinder is a solid shape characterised by two identical and parallel circular bases connected by a curved surface. When we refer to a cylinder in elementary geometry, we typically mean a right circular cylinder, where the line segment connecting the centres of the two circular bases (the axis) is perpendicular to the planes of the bases.

• It has 2 flat circular faces (the top and bottom bases).
• It has 1 curved lateral surface.
• It has 2 circular edges where the bases meet the curved surface.
• It has no vertices in the traditional sense (sharp corners).

Examples include soda cans, pipes, and circular pillars.

4. Cone (Right Circular Cone)

A cone is a solid shape that tapers smoothly from a flat base (typically circular) to a point called the apex or vertex. Similar to cylinders, the term usually refers to a right circular cone, where the apex is located directly above the centre of the circular base.

• It has 1 flat circular face (the base).
• It has 1 curved lateral surface.
• It has 1 circular edge (the boundary of the base).
• It has 1 vertex (the apex).

Examples include ice cream cones, traffic cones, and party hats.

5. Sphere

A sphere is a perfectly round solid shape in three-dimensional space. It is the 3D equivalent of a circle in 2D. A sphere is defined as the set of all points in space that are at a fixed distance (the radius) from a fixed point (the centre).

• It has only 1 continuous curved surface.
• It has no faces, edges, or vertices in the traditional sense.

Examples are ubiquitous: balls, marbles, and planets are all approximations of spheres.

6. Pyramid

A pyramid is a solid shape formed by connecting a polygonal base to a single point, called the apex. The connecting surfaces are triangular faces.

• It has 1 polygonal base face. The shape of the base determines the type of pyramid (e.g., a triangular base forms a triangular pyramid, a square base forms a square pyramid, a hexagonal base forms a hexagonal pyramid).
• It has triangular lateral faces (the number of triangular faces is equal to the number of sides of the base polygon).
• The edges include the edges of the base and the edges connecting the base vertices to the apex.
• The vertices include the vertices of the base and the single apex vertex.

A square pyramid, like the ancient Egyptian pyramids, has a square base and four triangular faces. It has 5 faces (1 square + 4 triangles), 8 edges, and 5 vertices (4 at the base + 1 apex).

A triangular pyramid (or tetrahedron) has a triangular base and three triangular faces. It is also considered a polyhedron with 4 faces, 6 edges, and 4 vertices.

A right pyramid is one where the apex is directly above the geometric centre of the base.

Another important category of solid shapes are prisms, which have two identical polygonal bases connected by rectangular or parallelogram faces. Examples include the rectangular prism (cuboid), triangular prism, pentagonal prism, etc.

Terms Related to Solid Shapes (Faces, Edges, Vertices)

When we describe and analyse the structure of solid shapes, particularly those with flat surfaces, we use specific terms to refer to their different parts. The most fundamental components are faces, edges, and vertices.

These terms are most clearly applicable to polyhedra, which are three-dimensional solid shapes whose boundaries are composed of flat polygonal faces.

Let's define these terms:

• Face: A face is a single, flat surface that forms part of the boundary of a solid shape. In the case of polyhedra, each face is a polygon (a closed figure with straight sides, like a square, triangle, or rectangle). Faces are the "sides" of the solid.

  Example: A standard dice (a cube) has six flat square faces.

• Edge: An edge is a line segment where two faces of a solid object meet. It is the boundary line shared by two adjacent faces.

  Example: A cube has 12 edges, which are the line segments forming the sides of its square faces.

• Vertex (plural: Vertices): A vertex is a point where three or more edges of a solid shape meet. Vertices are the "corners" of the solid.

  Example: A cube has 8 vertices, where three edges converge at each corner.

Faces, Edges, and Vertices for Common Polyhedra

We can count the number of faces (F), vertices (V), and edges (E) for various polyhedral shapes:

Euler's Formula for Polyhedra

For any convex polyhedron (a polyhedron where a line segment connecting any two points on its surface lies entirely within or on the surface), there exists a remarkable mathematical relationship between the number of its faces (F), vertices (V), and edges (E). This relationship was discovered by the famous mathematician Leonhard Euler and is known as Euler's Formula:

Let's verify Euler's Formula for some of the polyhedra from the table:

• For a Cube: $F=6$, $V=8$, $E=12$. $F + V - E = 6 + 8 - 12 = 14 - 12 = 2$. The formula holds true.
• For a Square Pyramid: $F=5$, $V=5$, $E=8$. $F + V - E = 5 + 5 - 8 = 10 - 8 = 2$. The formula holds true.
• For a Triangular Pyramid (Tetrahedron): $F=4$, $V=4$, $E=6$. $F + V - E = 4 + 4 - 6 = 8 - 6 = 2$. The formula holds true.
• For a Triangular Prism: $F=5$, $V=6$, $E=9$. $F + V - E = 5 + 6 - 9 = 11 - 9 = 2$. The formula holds true.

Euler's formula is a powerful result that applies to a wide class of polyhedra and provides a way to check the consistency of the counts of faces, vertices, and edges.

Terms for Curved Solids

For solid shapes that have curved surfaces, like cylinders, cones, and spheres, the terms "face," "edge," and "vertex" are sometimes used by extending the definitions, although they don't perfectly fit the strict definitions used for polyhedra.

• Cylinder: Commonly described as having 3 'faces' (the two flat circular bases and the curved lateral surface), 2 'edges' (the circular boundaries where the flat bases meet the curved surface), and 0 'vertices'.
• Cone: Commonly described as having 2 'faces' (the flat circular base and the curved lateral surface), 1 'edge' (the circular boundary of the base), and 1 'vertex' (the apex).
• Sphere: Described as having 1 continuous curved surface, 0 'edges', and 0 'vertices'.

It is important to be aware that these terms, when applied to curved solids, are sometimes used with slightly different interpretations than for polyhedra with flat surfaces only.

Answer:

A hexagonal prism has two hexagonal bases and rectangular faces connecting the corresponding sides of the bases.

• Faces (F): There are two hexagonal bases (top and bottom). Since a hexagon has 6 sides, there are 6 rectangular faces connecting the bases. Total faces = $2 (\text{hexagons}) + 6 (\text{rectangles}) = 8$. So, $F = 8$.
• Vertices (V): Each hexagonal base has 6 vertices. Since there are two bases, the total number of vertices is $6 (\text{on top base}) + 6 (\text{on bottom base}) = 12$. So, $V = 12$.
• Edges (E): There are 6 edges around the top base, 6 edges around the bottom base, and 6 edges connecting the vertices of the top base to the corresponding vertices of the bottom base. Total edges = $6 (\text{top base}) + 6 (\text{bottom base}) + 6 (\text{connecting}) = 18$. So, $E = 18$.

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

$F + V - E = 8 + 12 - 18$

$= 20 - 18$

$= 2$

The formula holds true for the hexagonal prism ($8 + 12 - 18 = 2$).