How to find the area of a trapezoid

A trapezoid’s slanted sides can make its area look tricky. We’ll use the average of the two parallel bases and the perpendicular height: $A=\frac{1}{2}(b_1+b_2)h$. Let’s work through two examples so you can see which measurements matter.

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Identify the two bases and perpendicular height

A trapezoid is a quadrilateral with exactly one pair of opposite parallel sides, using the definition in this lesson. Those parallel sides are the bases, $b_1$ and $b_2$. The height $h$ is the perpendicular distance between the bases.

The height joins the two parallel bases at right angles.

$A=\frac{1}{2}(b_1+b_2)h$

We add the base lengths, divide by two to find their average, then multiply by the height. Because we’re multiplying lengths, the result is measured in square units.

Example: Use the given dimensions

Find the area of the trapezoid. Its parallel bases have lengths $13$ and $10$, and the perpendicular height is $7$.

The trapezoid has bases 13 and 10 and height 7.

$A=\frac{1}{2}(b_1+b_2)h$

$A=\frac{1}{2}(13+10)\left(7\right)$

$A=\frac{1}{2}\left(23\right)\left(7\right)$

$A=80.5{\mathrm{units}}^2$

Example: Find the dimensions on a grid

Find the area of the trapezoid on the coordinate grid. We’ll use the grid to count the lengths of the parallel bases and the perpendicular distance between them.

A trapezoid drawn on a coordinate grid.

Counting the grid units gives base lengths of $11$ and $8$ and a height of $5$. The height is measured across the gap between the parallel bases, along a perpendicular line.

The grid shows bases 11 and 8 and perpendicular height 5.

$A=\frac{1}{2}(11+8)\left(5\right)$

$A=\frac{1}{2}\left(19\right)\left(5\right)$

$A=47.5{\mathrm{units}}^2$

The process is the same whether the dimensions are labeled or found from a grid: identify the two parallel bases and their perpendicular height, then substitute those three lengths into the area formula.

Related lesson: How to find the area of a parallelogram.

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