How to Find the Area of a Trapezoid

Grade: 7-8 | Topic: Geometry

What You Will Learn

After this lesson you will be able to calculate the area of any trapezoid using the standard formula. You will understand why the formula works, know how to identify the two bases and the height, and solve problems that involve trapezoids in real-world contexts such as land plots, cross-sections, and composite shapes.

Theory

What is a trapezoid?

A trapezoid (called a trapezium in British English) is a four-sided shape (quadrilateral) that has exactly one pair of parallel sides. The two parallel sides are called bases and are labeled $b_1$ and $b_2$. The perpendicular distance between the two bases is the height $h$.

The non-parallel sides are called legs. The legs can be equal (making it an isosceles trapezoid) or different lengths.

The area formula

The area of a trapezoid is:

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

This formula works because a trapezoid can be thought of as a shape whose width changes linearly from $b_1$ at one end to $b_2$ at the other. Taking the average of the two bases — $\frac{b_1 + b_2}{2}$ — gives the width of an equivalent rectangle that has the same area. Multiply that average width by the height and you get the area.

Another way to see it: duplicate the trapezoid, flip the copy upside-down, and place it next to the original. The two trapezoids form a parallelogram with base $(b_1 + b_2)$ and height $h$. The area of the parallelogram is $(b_1 + b_2) \times h$, so one trapezoid is half of that.

For example, a trapezoid with bases 6 cm and 10 cm and a height of 4 cm has area:

$A = \frac{1}{2}(6 + 10) \times 4 = \frac{1}{2}(16) \times 4 = 8 \times 4 = 32 \text{ cm}^2$

Special cases

• Parallelogram: If $b_1 = b_2$, the formula becomes $A = b \times h$ (the standard parallelogram formula).
• Triangle: If one base shrinks to zero ($b_2 = 0$), the formula becomes $A = \frac{1}{2} b_1 \times h$ (the triangle formula). A triangle is a "degenerate" trapezoid.

Identifying the height

The height $h$ is not the length of a leg. It is the perpendicular distance between the two parallel bases. In many diagrams the height is shown as a dashed line forming a right angle with both bases. If the problem gives you a leg length instead, you may need to use the Pythagorean theorem to find the height.

Worked Examples

Example 1: Straightforward calculation (easy)

Problem: A trapezoid has bases of 12 cm and 8 cm and a height of 5 cm. Find its area.

Step 1: Identify the values. $b_1 = 12$ cm, $b_2 = 8$ cm, $h = 5$ cm.

Step 2: Apply the formula. $A = \frac{1}{2}(12 + 8) \times 5 = \frac{1}{2}(20) \times 5 = 10 \times 5 = 50 \text{ cm}^2$

Answer: 50 cm$^2$

Example 2: Finding the area with decimal measurements (easy)

Problem: A trapezoid has bases of 7.5 m and 4.5 m and a height of 6 m. Find its area.

Step 1: Add the bases. $b_1 + b_2 = 7.5 + 4.5 = 12$

Step 2: Apply the formula. $A = \frac{1}{2}(12) \times 6 = 6 \times 6 = 36 \text{ m}^2$

Answer: 36 m$^2$

Example 3: Finding a missing base (medium)

Problem: A trapezoid has an area of 84 cm$^2$, one base of 10 cm, and a height of 7 cm. Find the other base.

Step 1: Start from the formula and solve for $b_2$. $A = \frac{1}{2}(b_1 + b_2) \times h$ $84 = \frac{1}{2}(10 + b_2) \times 7$

Step 2: Multiply both sides by 2. $168 = (10 + b_2) \times 7$

Step 3: Divide both sides by 7. $24 = 10 + b_2$

Step 4: Subtract 10. $b_2 = 14 \text{ cm}$

Answer: 14 cm

Example 4: Using the Pythagorean theorem to find the height (medium)

Problem: An isosceles trapezoid has bases of 20 cm and 12 cm, and each leg is 5 cm. Find the area.

Step 1: Find the horizontal distance each leg extends beyond the shorter base. $\text{overhang on each side} = \frac{20 - 12}{2} = \frac{8}{2} = 4 \text{ cm}$

Step 2: Use the Pythagorean theorem to find the height. Each leg (5 cm), the overhang (4 cm), and the height form a right triangle. $h^2 + 4^2 = 5^2 \implies h^2 = 25 - 16 = 9 \implies h = 3 \text{ cm}$

Step 3: Calculate the area. $A = \frac{1}{2}(20 + 12) \times 3 = \frac{1}{2}(32) \times 3 = 16 \times 3 = 48 \text{ cm}^2$

Answer: 48 cm$^2$

Example 5: Real-world cross-section (challenging)

Problem: A water channel has a trapezoidal cross-section. The bottom of the channel is 1.5 m wide, the top is 3.5 m wide, and the depth is 1.2 m. Find the area of the cross-section, then calculate the volume of water in a 50 m stretch of the channel when it is full.

Step 1: Find the area of the trapezoidal cross-section. $A = \frac{1}{2}(1.5 + 3.5) \times 1.2 = \frac{1}{2}(5) \times 1.2 = 2.5 \times 1.2 = 3 \text{ m}^2$

Step 2: Multiply by the length to get the volume. $V = A \times \text{length} = 3 \times 50 = 150 \text{ m}^3$

Answer: Cross-section area = 3 m$^2$, Volume = 150 m$^3$

Common Mistakes

Mistake 1: Using a leg length instead of the perpendicular height

❌ Bases are 10 and 6, leg is 5: $A = \frac{1}{2}(10 + 6) \times 5 = 40$

✅ The height must be perpendicular to both bases. If the actual perpendicular height is 4: $A = \frac{1}{2}(10 + 6) \times 4 = 32$

Why this matters: The leg is a slant side, not the height. Legs are longer than the perpendicular height (unless the legs are already vertical), so using a leg inflates the area.

Mistake 2: Forgetting to add the two bases before halving

❌ $A = \frac{1}{2} \times 10 \times 5 = 25$ (used only one base)

✅ $A = \frac{1}{2}(10 + 6) \times 5 = 40$ (used both bases)

Why this matters: Using only one base gives you the area of a triangle, not a trapezoid. The formula requires the sum of both parallel sides.

Mistake 3: Forgetting the one-half factor

❌ $A = (b_1 + b_2) \times h = (10 + 6) \times 5 = 80$

✅ $A = \frac{1}{2}(10 + 6) \times 5 = 40$

Why this matters: Without the $\frac{1}{2}$, you are computing the area of the full parallelogram formed by two copies of the trapezoid. Your answer will be exactly double the correct area.

Practice Problems

Try these on your own before checking the answers:

1. A trapezoid has bases of 9 cm and 15 cm and a height of 8 cm. Find its area.
2. A trapezoid has bases of 5.5 m and 2.5 m and a height of 4 m. Find its area.
3. The area of a trapezoid is 60 cm$^2$. One base is 7 cm and the height is 6 cm. Find the other base.
4. An isosceles trapezoid has bases of 16 m and 10 m, and each leg is 5 m. Find the area.
5. A flower bed is shaped like a trapezoid with parallel sides of 3 m and 5 m and a depth of 2 m. How many square metres of soil does it cover?

Click to see answers

1. $A = \frac{1}{2}(9 + 15) \times 8 = \frac{1}{2}(24) \times 8 = 96$ cm$^2$.
2. $A = \frac{1}{2}(5.5 + 2.5) \times 4 = \frac{1}{2}(8) \times 4 = 16$ m$^2$.
3. $60 = \frac{1}{2}(7 + b_2) \times 6 \implies 120 = (7 + b_2) \times 6 \implies 20 = 7 + b_2 \implies b_2 = 13$ cm.
4. Overhang: $\frac{16 - 10}{2} = 3$ m. Height: $h = \sqrt{5^2 - 3^2} = \sqrt{16} = 4$ m. Area: $\frac{1}{2}(16 + 10) \times 4 = 52$ m$^2$.
5. $A = \frac{1}{2}(3 + 5) \times 2 = 8$ m$^2$.

Summary

• A trapezoid has exactly one pair of parallel sides called bases ($b_1$ and $b_2$).
• The area formula is $A = \frac{1}{2}(b_1 + b_2) \times h$, where $h$ is the perpendicular height between the bases.
• The formula averages the two bases and multiplies by the height — this is equivalent to half a parallelogram.
• If only leg lengths are given (not the height), use the Pythagorean theorem to find $h$.
• The formula reduces to the parallelogram formula when $b_1 = b_2$ and to the triangle formula when one base is zero.

Related Topics

• Area and Perimeter — Formulas, Examples, and Practice
• Area of a Triangle — Formula and Examples
• How to Find the Perimeter of Any Polygon

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