# Area of a trapezoid - derivation

This page describes how to derive the forumula for the area of a trapezoid by creating a parallelogram from two congruent trapezoids. The formula is simply one half the area of this parallelogram.

1. Start with a trapezoid with known base lengths (b1, b2) and altitude (height).
2. Make a copy of it.
3. Rotate the copy 180°.
4. Translate (move) the copy to touch the original.

Click on the "Run" button below to see this in action.

When the two trapezoids are combined in this way, the result is a parallelogram, which has two pairs of opposite, congruent sides.

Recall that the area of a parallelogram is its altitude (h) times the length of either base. From the figure above we see that both base lengths are equal to b1+b2. So the area of the parallelogram is Since this is the area of two trapezoids we have to divide this by two, giving

## Finally..

This can be rearranged into more familar forms: or

While you are here..

... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone. However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site? When we reach the goal I will remove all advertising from the site.

It only takes a minute and any amount would be greatly appreciated. Thank you for considering it!   – John Page

Become a patron of the site at   patreon.com/mathopenref