Area of a Trapezoid — Formula & Step-by-Step

A trapezoid has one pair of parallel sides, and its area is the average of those two parallel sides times the perpendicular height. This method works for any trapezoid as long as you can identify the two parallel sides and the true height between them.

Here, $b_1$ and $b_2$ are the two parallel sides, and $h$ is the perpendicular height between them. A slanted side is not the height unless it meets the bases at a right angle.

When to use this method

Reach for the trapezoid area formula whenever a four-sided figure has exactly one pair of parallel sides. It appears in geometry class, composite-shape problems, floor plans, land-measurement diagrams, and coordinate geometry when a quadrilateral has one pair of parallel sides. The whole task usually comes down to choosing the correct pair of parallel sides and the genuine perpendicular height.

Step by step

1. Find the parallel sides. Identify the two sides that are parallel. These are the base lengths $b_1$ and $b_2$ used in the formula.
2. Find the height. Use the perpendicular distance between the parallel sides, not a slanted side length.
3. Apply the formula. Substitute into $A = \frac{1}{2}(b_1 + b_2)h$.
4. Simplify carefully. Add the bases first, then multiply by the height, then divide by $2$.
5. Check the units. Area is written in square units such as $\text{cm}^2$ or $\text{m}^2$.

A useful way to read the formula: a trapezoid acts like a rectangle whose width is the average of the two parallel sides, since $A = \left(\frac{b_1 + b_2}{2}\right)h$. If the parallel sides were equal, the trapezoid would become a rectangle and the formula would reduce to $A = \frac{1}{2}(b + b)h = bh$, a quick sanity check.

A full worked example

Take parallel sides of $8$ cm and $14$ cm and a perpendicular height of $5$ cm. Substitute into the formula:

So the area is $55\ \text{cm}^2$. As a check, the average of $8$ and $14$ is $11$, so the trapezoid matches a rectangle with width $11$ cm and height $5$ cm, which also gives $55\ \text{cm}^2$.

Where students get stuck, and how to check each step

• Picking the bases (step 1): Using a non-parallel side as a base is the most frequent slip. Self-check that the two chosen sides are actually parallel.
• Reading the height (step 2): A slanted side is not the height unless it is perpendicular; a marked right angle points to your $h$.
• Applying the formula (step 3-4): Forgetting the factor of $\frac{1}{2}$, or multiplying only one base by the height instead of both parallel sides, changes the answer.
• Final units (step 5): A plain unit instead of a square unit is a structure error.

To practice, set up parallel sides $6$ m and $10$ m with height $4$ m and run all five steps, then change only the height to see how the area scales.