# 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.

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.

Observe that $A_1 = (ah)/2$$A_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

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

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