Area of Basic Plane Figures

Area of Basic Plane Figures

Area of Squares and Rectangles

The area of a plane figure is the measure of the surface enclosed by its boundary. It quantifies the amount of flat space that a two-dimensional shape occupies. For squares and rectangles, the area is calculated based on the lengths of their sides.

Area of a Rectangle

A rectangle is a quadrilateral with four right angles ($90^\circ$) and opposite sides equal in length. The longer side is usually called the length ($l$), and the shorter side is called the width or breadth ($w$).

Conceptually, the area of a rectangle represents the total number of unit squares (squares with side length 1 unit) that can fit perfectly inside its boundary without overlapping. If a rectangle has a length of '$l$' units and a width of '$w$' units, you can arrange '$l$' unit squares along the length and '$w$' unit squares along the width. The total count of these unit squares that fill the rectangle is the product of the number of squares along the length and the number of squares along the width.

Formula:

Based on this concept, the formula for the area of a rectangle ($A_{\text{rectangle}}$) is:

The unit of area is derived from the unit of length used for the sides. If $l$ and $w$ are in centimetres (cm), the area will be in square centimetres ($\text{cm}^2$). If they are in metres (m), the area will be in square metres ($\text{m}^2$), and so on. The unit is always the square of the linear unit.

Answer:

Area of a Square

A square is a special type of rectangle where all four sides are equal in length, and all four angles are right angles ($90^\circ$).

Let the length of each side of the square be '$s$'. Since a square is a rectangle with its length equal to its width ($l = s$ and $w = s$), we can use the rectangle area formula to find the area of a square.

Derivation of the Formula:

Using the formula for the area of a rectangle (2):

Substitute $l=s$ and $w=s$ into the formula:

Formula:

The formula for the area of a square ($A_{\text{square}}$) is:

$\textbf{Area of Square} = \mathbf{(Side \$\$ Length)^2}$

Or simply, $\mathbf{A = s^2}$.

Similar to the rectangle, the unit of area for a square is the square of the unit used for the side length (e.g., $\text{cm}^2$, $\text{m}^2$, $\text{km}^2$).

Answer:

Area of Basic Plane Figures

Area of a Parallelogram

A parallelogram is a quadrilateral where both pairs of opposite sides are parallel and equal in length. Unlike a rectangle or square, the angles of a parallelogram are generally not $90^\circ$ (unless it's a rectangle or square itself).

To calculate the area of a parallelogram, we need two specific measurements:

• The base ($b$): This can be the length of any one side of the parallelogram.
• The corresponding height ($h$): This is the perpendicular distance from the side opposite the chosen base to the base itself. It is crucial that the height measurement forms a right angle ($90^\circ$) with the base. The height is often represented by a dashed line drawn inside or outside the parallelogram.

Formula and Derivation

The area of a parallelogram is given by the product of its base and its corresponding height.

Formula:

Derivation (Conceptual Explanation):

The formula for the area of a parallelogram can be understood by relating it to the area of a rectangle. Consider a parallelogram ABCD:

Draw a perpendicular from vertex D to the base AB, meeting AB at point E. This segment DE represents the height ($h$) corresponding to the base AB ($b$).

Now, imagine cutting off the right-angled triangle ADE from the parallelogram. The remaining shape is a trapezium EBCD.

If you slide the triangle ADE and attach it to the side BC such that AD coincides with BC (which is possible because opposite sides are equal and parallel), the point D will coincide with C, and the point E will move to a new position, say F, on the extension of BC such that BF is perpendicular to the extension. The resulting figure is the rectangle EFCD.

The base of this rectangle EF is equal to the base AB of the parallelogram, which is '$b$'. The height of this rectangle FC is equal to the perpendicular height DE of the parallelogram, which is '$h$'.

The area of the parallelogram ABCD is equal to the area of the rectangle EFCD because we have merely rearranged the parts without adding or removing any area.

Area of Rectangle EFCD = Length $\times$ Width = EF $\times$ FC = $b \times h$.

Therefore,

This derivation visually demonstrates why the area of a parallelogram is calculated by multiplying its base by its corresponding perpendicular height.

Important Note: The height must be the perpendicular distance to the base. Do not use the length of the slanted side unless the figure is a rectangle (where the slanted side is also the height).

Examples

Answer:

Answer:

Area of a Triangle (using Base and Height)

A triangle is the simplest polygon, a closed two-dimensional shape with three sides and three vertices. The area of a triangle is the amount of surface enclosed by its three sides.

To calculate the area of a triangle using the base and height method, we need to identify:

• The base ($b$): This can be any one of the three sides of the triangle. The choice of base will determine the corresponding height.
• The corresponding height ($h$): This is the perpendicular distance from the vertex opposite the chosen base to the base itself (or the line containing the base, if the vertex is outside the base segment). It must form a right angle ($90^\circ$) with the base line.

It's important to understand that the height's position relative to the base depends on the type of triangle:

• In an acute triangle (all angles less than $90^\circ$), the height corresponding to any base falls inside the triangle.
• In a right-angled triangle, if one of the legs (the sides forming the right angle) is chosen as the base, the other leg is the corresponding height. The height falls along one of the sides.
• In an obtuse triangle (one angle greater than $90^\circ$), the height corresponding to a base adjacent to the obtuse angle falls outside the triangle. In such cases, the base line must be extended to draw the perpendicular height from the opposite vertex.

Formula and Derivation

The area of a triangle is half the product of its base and its corresponding height.

Formula:

Derivation:

The formula for the area of a triangle can be derived by observing its relationship with a parallelogram.

Consider any triangle, say $\triangle \text{ABC}$, with base BC of length '$b$' and corresponding height AD of length '$h$', where AD is perpendicular to BC.

Imagine creating an identical copy of $\triangle \text{ABC}$. Rotate this copy by $180^\circ$ and place it adjacent to the original triangle along one of its sides (say, AC). When placed together appropriately, the two identical triangles will form a parallelogram.

The base of this parallelogram will be the base of the original triangle, which is '$b$'. The height of this parallelogram will be the perpendicular distance between the parallel bases, which is the height of the original triangle, '$h$'.

The area of this parallelogram is given by the formula: Area = Base $\times$ Height $= b \times h$.

Since the parallelogram is formed by two congruent (identical) triangles, the area of the original triangle must be exactly half the area of the parallelogram.

This derivation shows that any triangle's area is directly related to the area of a parallelogram with the same base and height, justifying the formula $A = \frac{1}{2} b h$.

Examples

Answer:

Answer:

Given:

Shape is a right-angled triangle.

Area, $A = 50 \$ \text{m}^2$.

Length of one side containing the right angle $= 10$ m.

To Find:

The length of the other side containing the right angle.

Solution:

In a right-angled triangle, the two sides that form the right angle are perpendicular to each other. This means we can consider one of these sides as the base and the other as the corresponding height.

Let the base ($b$) be the side with length $10$ m. Then the height ($h$) corresponding to this base is the length of the other side containing the right angle. We are given the Area and need to find $h$.

The formula for the area of a triangle is $A = \frac{1}{2} b h$. Using formula (2):

$\text{Area} = \frac{1}{2} b h$

Substitute the given area and base:

$50 \$ \text{m}^2 = \frac{1}{2} \times 10 \$ \text{m} \times h$

[Substituting $A=50$ $\text{m}^2$ and $b=10$ m]

$50 \$ \text{m}^2 = 5 \$ \text{m} \times h$

[Simplifying $\frac{1}{2} \times 10$]

Now, rearrange the equation to solve for $h$. Divide both sides by $5$ m:

$\frac{50 \$ \text{m}^2}{5 \$ \text{m}} = h$

$\mathbf{h = 10 \$\$ m}$

Therefore, the length of the other side containing the right angle is 10 metres (m).

Area of Basic Plane Figures

Area of Trapezium (Implicit/using breakdown)

A trapezium (also known as a trapezoid in some regions) is a quadrilateral defined by having at least one pair of parallel sides. These parallel sides are called the bases of the trapezium. The perpendicular distance between the parallel sides is called the height ($h$) of the trapezium.

Let the lengths of the two parallel sides be '$a$' and '$b$', and let the height (the perpendicular distance between them) be '$h$'.

Formula Derivation (Using Breakdown)

We can derive the formula for the area of a trapezium by breaking it down into simpler shapes whose area formulas we already know (like triangles and rectangles), or by relating it to a parallelogram.

Derivation Method 1: Breaking into a Rectangle and Triangles

Consider a trapezium ABCD where AB is parallel to DC. Let the parallel sides have lengths $a = \text{DC}$ and $b = \text{AB}$. Let the height be $h$. Draw perpendiculars from A and B to the line containing DC, meeting at points X and Y respectively. AX and BY are the heights, so AX = BY = $h$.

The trapezium is divided into $\triangle \text{ADX}$, rectangle ABYX (if X and Y are on DC, implying a right trapezium), or potentially triangles and a rectangle in between. For a general trapezium, if we drop perpendiculars from the endpoints of the shorter base (say, AB) to the longer base (say, DC), we divide the longer base into three segments. Let the lengths of these segments be $x, y,$ and $b$. The segments $x$ and $y$ form the bases of two right-angled triangles, and the segment of length $b$ forms the side of a rectangle with height $h$. The total length of the longer base is $a = x + b + y$.

The area of the trapezium is the sum of the areas of the shapes it is divided into:

Factor out the common term $h$:

We can rewrite the expression inside the parenthesis using the relation $a = x + b + y$. Note that $x + 2b + y = (x + b + y) + b = a + b$.

Substitute this back into the area equation:

Derivation Method 2: Breaking into Two Triangles using a Diagonal

Consider the same trapezium ABCD with AB parallel to DC, lengths $a$ and $b$, and height $h$. Draw a diagonal, say AC. This diagonal divides the trapezium into two triangles: $\triangle \text{ADC}$ and $\triangle \text{ABC}$.

The area of the trapezium is the sum of the areas of these two triangles:

Using the formula for the area of a triangle ($A = \frac{1}{2} \times \text{base} \times \text{height}$):

• For $\triangle \text{ADC}$: The base can be taken as DC, which has length $a$. The height corresponding to this base is the perpendicular distance from vertex A to the line DC. Since AB is parallel to DC and the height of the trapezium is the perpendicular distance between the parallel lines, this height is $h$.

  Area($\triangle \text{ADC}$) $= \frac{1}{2} \times \text{DC} \times h = \frac{1}{2} \times a \times h$

• For $\triangle \text{ABC}$: The base can be taken as AB, which has length $b$. The height corresponding to this base is the perpendicular distance from vertex C to the line AB. Since AB is parallel to DC, the perpendicular distance between AB and the line containing C (which is line DC) is also $h$.

  Area($\triangle \text{ABC}$) $= \frac{1}{2} \times \text{AB} \times h = \frac{1}{2} \times b \times h$

Summing the areas of the two triangles:

Factor out the common term $\frac{1}{2} h$:

Formula

Both derivation methods confirm the formula for the area of a trapezium:

where '$a$' and '$b$' are the lengths of the parallel sides, and '$h$' is the perpendicular distance between them.

Example

Answer: