Making Connection Between Areas of Trapezoids and Parallelograms

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 b_1 and b_2 and altitude h.

area of trapezoid

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.

Area of Trapezoids and Parallelograms

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

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

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

And we are done.

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