Derivation of the Area of a Trapezoid

A trapezoid (or trapezoid) is a quadrilateral with exactly one pair* of parallel sides. In the figure below, ABCD is a trapezoid and AB is parallel to CD.

area of a trapezoid

In this post, we derive the area of a trapezoid. We use the fact that a trapezoid can be partitioned into two triangles and one rectangle. The area A of a trapezoid is equal to the sum of the areas A_1, A_3 of the two triangles and the area A_2 of the rectangle.

area of a trapezoid2

Observe that A_1 = (ah)/2A_2 = b_1h, and A_3 = (ch)/2.

We know that

area of trapezoid  = area of triangle 1 + area of rectangle + area of triangle 2.

which means that A = A_1 + A_2 + A_3.

Substituting the values we have

\displaystyle \frac{ah}{2} + bh + \frac{ch}{2} =\displaystyle \frac{ah + 2b_1h + ch}{2}.

Simplifying the equation, rearranging the terms, and factoring result to

A = \displaystyle\frac{h}{2} [b_1 + (a + b_1 + h)].

If we let the longer base of the trapezoid be b_2, then, b_2 = a + b_1 + h. Substituting we have

A =\displaystyle \frac{h}{2}\left ( b_1 + b_2 \right ).

Therefore the area of a trapezoid A with base b_1 and b_2 and altitude h is

A = \displaystyle \frac{h}{2} \left ( b_1 + b_2 \right )


*Other books define trapezoid as a quadrilateral with at least 1 pair of parallel sides.

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