# A trapezoid has a top base of 32 and a bottom base of 22. Its legs are each 13. What is its area?

Nov 15, 2015

$324$

#### Explanation:

The area of a trapezoid is the average of its bases times its height. This can be expressed as ${A}_{\text{trapezoid}} = \frac{h \left({b}_{1} + {b}_{2}\right)}{2}$.

We have the two bases, but we don't have the height. Draw the trapezoid on a piece of paper. Now, write in the measurements. If both legs are $13$, and the bases are $22$ and $32$, how can we find the height?

We can use the Pythagorean theorem. If one base is $22$ and the other is $32$, the long base is $10$ longer than the short one. If we have an isosceles trapezoid like in the picture, the long base will extend $5$ more in either direction. So, if you drew a line cutting straight down from the vertex where the SHORT base and the height met all the way to the LONG base, you will have a right triangle with a hypotenuse of $13$ and a leg of $5$.

We can use the Pythagorean theorem to figure out that the other leg of the right triangle, which is also the height of the trapezoid, has length $12$.

Then, we plug our height and base lengths back into the area formula to determine that the area of the trapezoid is color(red)(324.