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 * 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 * 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 * w * 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