Area of a Parallelogram

Sub Topics
Let us first define a parallelogram, a two dimensional shape can be have any number of sides starting from three and hence they are called polygons. The general name for a four sided polygon is called a quadrilateral. The word 'quadrilateral' is derived from Latin meaning a four sided figure. As a special type of a quadrilateral, the shape may be having two sets of parallel and congruent sides. Such a specialized quadrilateral is known as 'parallelogram' highlighting the fact that the two sets of sides (the two opposite sides) are parallel. Thus, a parallelogram is a quadrilateral in which the opposite sides are parallel and congruent.

Now let us discuss about area. Area of a shape is the measure of the surface covered by that. Hence area of a parallelogram is the surface contained by a parallelogram.

Formula for Area of a Parallelogram

The area of a parallelogram can be found by using several formulas which are derived as explained earlier. The most common formula for the area A of a parallelogram is,

A = b $\times$  h, where 'b' is the base and 'h' is the vertical height of the parallelogram.
Let us see how this formula has been derived from fundamentals.

Let ABCD is the given parallelogram. Extend the line segment AB to AF such that CF is perpendicular to AF. Draw the perpendicular DE on AF from D. It could be seen that DECF is a rectangle.

Considering $\triangle$ AED and BFC,

$\angle$ AED = $\angle$ BFC (both are perpendiculars to parallel lines)

AD = BC (opposite sides of parallelogram)

DE = CF (distance between same parallel lines)

Hence, $\triangle$ AED $\equiv$ $\triangle$ BFC

Now, the area of the parallelogram ABCD = area of rectangle DEFC + area of triangle - area of triangle BFC = area of rectangle DEFC (since $\triangle$ AED $\equiv$ $\triangle$ BFC)
= b $\times$ h

In many cases, the measurement of vertical height of a parallelogram may not be given or may not even be possible. Also, a parallelogram is more commonly specified by its sides. On the other hand it will be easier to measure the angle between the sides, like angle A in the above figure. (Even if angle B is given, angle A can be figured out.

Since A is the supplementary of angle B). Hence, we can use the trigonometry and represent the vertical height of the parallelogram as

'w * sin x' where 'w' is the remaining side and 'x' is the angle between the sides 'b' and 'w'. Thus, the area A of the triangle can also expressed as,

A = b $\times$ w  $\times$ sin x

How to Find the Area of a Parallelogram

Area of parallelogram can be determined by different methods, depending upon the available parameters. Basically a parallelogram is imagined to be a combination of two congruent triangles. Hence the area of the parallelogram can be found by modifying the triangle area formula (just doubling the right side). If in case, the measures one of the diagonals and measures of both sides are known, then the area of each triangle can be determined by using Heron's formula.

Another concept is to geometrically construct to show that the area of the parallelogram is the same as the area of a rectangle, having its length and height same as the base and height of the parallelogram respectively. Sometimes, the sides of the parallelogram and the angle between them may be given. In that case, trigonometric concept may be used to determine the vertical concept and in turn the area of the parallelogram can be figured out.

Below you could see solved examples for area of parallelogram

Example 1.

Find the area of a parallelogram with a base of 20 cm and a height of 10 cm ?

Solution :

Given base=20 cm
Height=10 cm
Area=base $\times$ height
=>Area = 20 $\times$ 10
=>Area = 200 cm2

Example 2.
The area of a parallelogram is 125 cm2 and the height is 25 cm .find the base of the parallelogram?

Solution:
Given Area= 125 cm2
and height =25 cm
base=?

We have
Area = base $\times$ height
=>120 = base $\times$ 25
=>base = 125/25
=>base = 5 cm

Example 3.
The area of a parallelogram is 230 cm2 and the base is 10 cm .find the height of the parallelogram?

Solution:

Given Area= 230 cm2
and base=10 cm
height =?

We have
Area=base $\times$ height
=>230=10 $\times$ height
=>height = 230/10
=>height = 23 cm