Area of Trapezoids

Learn how to calculate the area of trapezoids using the formula with parallel sides and height.

Intermediate20 minLesson

Definition

A trapezoid is a quadrilateral with exactly one pair of parallel sides. These parallel sides are called the bases (

b_1

and

b_2

The area of a trapezoid is found using the formula:

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

Where:

$b_1$ = length of the first base (one parallel side)
$b_2$ = length of the second base (other parallel side)
$h$ = height (perpendicular distance between the bases)

You can also write this as:

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

The formula finds the average of the two bases, then multiplies by the height.

Try it now

What is the formula for the area of a trapezoid?

Worked Examples

Find the area of a trapezoid with bases of 8 cm and 12 cm, and a height of 5 cm.

Write the formula

$A = \frac{(b_1 + b_2) \times h}{2}$ → Area formula identified

Identify the values

$b_1 = 8$ cm, $b_2 = 12$ cm, $h = 5$ cm → Values identified

Add the bases

$b_1 + b_2 = 8 + 12 = 20$ cm → Sum of bases = 20 cm

Multiply by height

$20 \times 5 = 100$

Divide by 2

$\frac{100}{2} = 50$ → 50 cm²

Answer: The area is $50$ cm²

Common Mistakes

Forgetting to divide by 2

Why it's wrong: The formula requires dividing by 2 because a trapezoid is half of a parallelogram formed by the sum of the bases.

Correct: Always include the division: $A = \frac{(b_1 + b_2) \times h}{2}$ , not $(b_1 + b_2) \times h$

Using a slant side instead of the height

Why it's wrong: The height must be perpendicular to both bases. The slanted sides are not the height.

Correct: Always use the perpendicular distance between the parallel sides as the height.

Multiplying the bases instead of adding them

Why it's wrong: The formula requires adding $b_1 + b_2$ , not multiplying.

Correct: Remember: ADD the bases first, then multiply by height, then divide by 2.

Interactive Visual

Area and perimeter

Base b₁10Base b₂6Height h4

Area32

$A = \dfrac{(b_1 + b_2) \cdot h}{2} = \dfrac{(10 + 6) \cdot 4}{2} = 32$

Perimeter24.94

$P = b_1 + b_2 + 2c \approx 24.94$

Take the average of the two bases, then multiply by the height (drawn isosceles here).

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Practice Problems

17 problems

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What is the formula for the area of a trapezoid?

Why It Matters

Trapezoid area calculations appear in many real-world situations:

Architecture: Cross-sections of roofs, bridges, and ramps are often trapezoidal
Construction: Retaining walls, dam faces, and channel cross-sections
Land surveying: Irregularly shaped plots often contain trapezoidal sections
Engineering: Many structural components use trapezoidal shapes for stability
Art and design: Creating balanced compositions with non-rectangular shapes

Understanding trapezoid area helps you work with any four-sided shape that has parallel sides!

Real World Applications

Roof Cross-Sections

Many roof designs have trapezoidal cross-sections where calculating area helps determine material needs.

Example:

A roof section has parallel edges of 6 m (top) and 10 m (bottom), with a height of 2.5 m. Area = $\frac{(6 + 10) \times 2.5}{2} = 20$ m²

1Try It Yourself

A shed roof has parallel edges of 4 m and 7 m, with a height of 2 m.

What is the cross-sectional area of the roof?

Step 1: Write the mathematical expression

Calculate: $\frac{(4 + 7) \times 2}{2}$

Swimming Pool Design

Many swimming pools have a trapezoidal shape when viewed from above to fit irregular yard spaces.

Example:

A pool is 8 m at one end, 12 m at the other, and 15 m long (perpendicular distance). Area = $\frac{(8 + 12) \times 15}{2} = 150$ m²

2Try It Yourself

A decorative pond has widths of 3 m and 5 m at opposite ends, with a length of 6 m between them.

What is the surface area of the pond?

Step 1: Write the mathematical expression

Calculate: $\frac{(3 + 5) \times 6}{2}$

Key Takeaways

1A trapezoid has exactly one pair of parallel sides called bases ( $b_1$ and $b_2$ )
2The area formula is $A = \frac{(b_1 + b_2) \times h}{2}$
3The height ( $h$ ) must be perpendicular to both bases
4The formula works by finding the average of the two bases, then multiplying by the height
5Always remember to divide by 2 at the end

Frequently Asked Questions

Adding the bases and dividing by 2 gives you the average length of the bases. A trapezoid's area equals this average base times the height, which is why the formula works.

Glossary

Trapezoid: A quadrilateral with exactly one pair of parallel sides
Bases: The two parallel sides of a trapezoid, labeled $b_1$ and $b_2$
Height: The perpendicular distance between the two bases of a trapezoid
Parallel sides: Sides that never intersect and remain the same distance apart

Formula Card

Area of a Trapezoid

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

Add the two bases, multiply by the height, then divide by 2. Variables: $b_1$ and $b_2$ are the parallel sides, $h$ is the perpendicular height.