Area of a Trapezoid

A trapezoid (also called a trapezium) is a four-sided polygon with exactly one pair of parallel sides called the bases. Finding its area requires one formula that averages the two bases, then multiplies by the height. This is a standard GED geometry topic that appears in both computation and word-problem formats.

What Is a Trapezoid?

A trapezoid is a quadrilateral with exactly one pair of parallel sides. The two parallel sides are called the bases (b₁ and b₂). The perpendicular distance between the bases is the height (h). The non-parallel sides are called the legs.

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Area Formula

A = ½ × \(\color{blue}{(b<\text{ sub }>1 + b<\text{ sub }>2) \times h}\)

• b₁ = length of the first (shorter or longer) base
• b₂ = length of the second base
• h = perpendicular height between the two bases

Why does this work? A trapezoid can be split into two triangles, each sharing the height. Adding their areas gives \(\color{blue}{\frac{1}{2}b_{1}h + \frac{1}{2}b_{2}h = \frac{1}{2}(b_{1} + b_{2})h}\).

How to Use the Formula

Identifying bases and height

The two bases are the parallel sides. The height is always measured perpendicular to the bases — not along the slanted legs.

Applying the formula

1. Add the two bases: \(\color{blue}{b_{1} + b_{2}}\).
2. Multiply by the height: \(\color{blue}{(b_{1} + b_{2}) \times h}\).
3. Divide by 2: \(\color{blue}{\frac{1}{2} \times (b_{1} + b_{2}) \times h}\).

Step-by-Step Summary

1. Identify the two parallel bases and the perpendicular height.
2. Add the two bases.
3. Multiply the sum by the height.
4. Divide by 2 and label with square units.

Watch: Area of a Trapezoid (Video Lesson)

Math with Mr. J demonstrates the trapezoid area formula with clear, step-by-step examples:

Worked Examples

Example 1: Find the area of a trapezoid with bases 4 cm and 8 cm and height 5 cm.

\(\color{blue}{A = \frac{1}{2} \times (4 + 8) \times 5 = \frac{1}{2} \times 12 \times 5 = \frac{1}{2} \times 60 = 30 \text{ cm }^{2}}\)

Example 2: A trapezoid has bases 6 m and 10 m and a height of 4 m. Find the area.

\(\color{blue}{A = \frac{1}{2} \times (6 + 10) \times 4 = \frac{1}{2} \times 16 \times 4 = \frac{1}{2} \times 64 = 32 m^{2}}\)

Example 3: A trapezoidal garden has parallel sides of 3 yd and 7 yd with a perpendicular height of 6 yd. How many square yards of grass are needed?

\(\color{blue}{A = \frac{1}{2} \times (3 + 7) \times 6 = \frac{1}{2} \times 10 \times 6 = \frac{1}{2} \times 60 = 30 \text{ yd }^{2}}\)

Example 4: The area of a trapezoid is 21 ft². The height is 3 ft and one base is 5 ft. Find the other base.

\(\color{blue}{21 = \frac{1}{2} \times (5 + b_{2}) \times 3 \rightarrow 21 = 1.5(5 + b_{2}) \rightarrow 14 = 5 + b_{2} \rightarrow b_{2} = 9 \text{ ft }}\)

More Practice: Area of a Trapezoid (Video)

Khan Academy explains the formula and works through additional geometry examples:

Exercises

1. Find the area of a trapezoid with bases 5 in. and 9 in. and height 3 in.
2. A trapezoid has bases 7 cm and 13 cm and height 6 cm. Find the area.
3. A trapezoidal roof section has parallel edges of 10 ft and 16 ft, with a perpendicular height of 8 ft. Find the area.
4. The area of a trapezoid is 45 cm². The two bases are 6 cm and 9 cm. Find the height.
5. A trapezoid has equal bases of 8 m each and height 5 m. What shape is it effectively, and what is the area?
6. Bases are 4 ft and 10 ft; height is 7 ft. Find the area.

Answers

1. \(\color{blue}{\frac{1}{2}(5+9)(3) = \frac{1}{2}\times 12\times 3 = 18 \text{ in }^{2}}\)
2. \(\color{blue}{\frac{1}{2}(7+13)(6) = \frac{1}{2}\times 20\times 6 = 60 \text{ cm }^{2}}\)
3. \(\color{blue}{\frac{1}{2}(10+16)(8) = \frac{1}{2}\times 26\times 8 = 104 \text{ ft }^{2}}\)
4. \(\color{blue}{45 = \frac{1}{2}(6+9)h \rightarrow 45 = 7.5h \rightarrow h = 6 \text{ cm }}\)
5. Equal bases mean it is a parallelogram (rectangle); \(\color{blue}{A = 8\times 5 = 40 m^{2}}\) (also confirmed by formula: ½\(\color{blue}{(8+8)(5) = 40}\))
6. \(\color{blue}{\frac{1}{2}(4+10)(7) = \frac{1}{2}\times 14\times 7 = 49 \text{ ft }^{2}}\)

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Frequently Asked Questions

What is the difference between height and slant height in a trapezoid?

The height is the perpendicular distance between the two parallel bases — always used in the area formula. The slant height is the length of a non-parallel leg. Do not confuse them.

Is a parallelogram a special trapezoid?

Yes — a parallelogram has two pairs of parallel sides, which means it qualifies as a trapezoid (by the inclusive definition). If you use \(\color{blue}{A = \frac{1}{2}(b_{1}+b_{2})h}\) with \(\color{blue}{b_{1} = b_{2}}\), you get \(\color{blue}{A = b\times h}\), which is the parallelogram formula.

What if the trapezoid is right-angled?

A right trapezoid has one leg perpendicular to the bases. This makes the height equal to that leg’s length, so you can read it directly from the figure. The area formula is the same.

Related Topics

• Area of a Parallelogram
• Area of a Triangle
• Polygons
• Finding Area of Compound Figures
• Triangles