Our last two discussions was about deriving the areas of trapezoids and parallelograms. In this post, we relate the two areas. We derive the area of a trapezoid using the area of a parallelogram. In the following derivation, we use the trapezoid with bases $latex b_1$ and $latex b_2$ and altitude $latex h$.

To form a parallelogram using a trapezoid, make a copy of a trapezoid and then rotate it 180 degrees and make the corresponding sides coincide as shown below.

Recall that the area of the parallelogram is the product of its base and its altitude, therefore the area $latex A_p$ of the parallelogram in the figure above is

$latex A_p = h(b_1 + b_2)$.

Since the area of the parallelogram above is twice the area of the trapezoid, we divide it by 2 making the area of the trapezoid

$latex A = \displaystyle \frac{h}{2} (b_1 + b_2)$.

And we are done.