Area of Trapezoids

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

Intermediate20 minLesson
Printable worksheet

Definition

A trapezoid is a quadrilateral with exactly one pair of parallel sides. These parallel sides are called the bases (b1b_1 and b2b_2).
The area of a trapezoid is found using the formula:
A=(b1+b2)×h2A = \frac{(b_1 + b_2) \times h}{2}
Where:
  • b1b_1 = length of the first base (one parallel side)
  • b2b_2 = length of the second base (other parallel side)
  • hh = height (perpendicular distance between the bases)
You can also write this as:
A=12(b1+b2)×hA = \frac{1}{2}(b_1 + b_2) \times h
The formula finds the average of the two bases, then multiplies by the height.

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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.
1

Write the formula

A=(b1+b2)×h2A = \frac{(b_1 + b_2) \times h}{2} → Area formula identified
2

Identify the values

b1=8b_1 = 8 cm, b2=12b_2 = 12 cm, h=5h = 5 cm → Values identified
3

Add the bases

b1+b2=8+12=20b_1 + b_2 = 8 + 12 = 20 cm → Sum of bases = 20 cm
4

Multiply by height

20×5=10020 \times 5 = 100
5

Divide by 2

1002=50\frac{100}{2} = 50 → 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=(b1+b2)×h2A = \frac{(b_1 + b_2) \times h}{2}, not (b1+b2)×h(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 b1+b2b_1 + b_2, not multiplying.

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

Interactive Visual

Area and perimeter

b₁ = 10b₂ = 6h = 4

Area32

A=(b1+b2)⋅h2=(10+6)⋅42=32A = \dfrac{(b_1 + b_2) \cdot h}{2} = \dfrac{(10 + 6) \cdot 4}{2} = 32

Perimeter24.94

P=b1+b2+2c≈24.94P = 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

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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 = (6+10)×2.52=20\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: (4+7)×22\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 = (8+12)×152=150\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: (3+5)×62\frac{(3 + 5) \times 6}{2}

Key Takeaways

  • 1A trapezoid has exactly one pair of parallel sides called bases (b1b_1 and b2b_2)
  • 2The area formula is A=(b1+b2)×h2A = \frac{(b_1 + b_2) \times h}{2}
  • 3The height (hh) 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.
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.
It doesn't matter which base you call b1b_1 or b2b_2. Since we add them together, the order doesn't affect the result.
If both bases are equal (b1=b2b_1 = b_2), the trapezoid becomes a rectangle, and the formula simplifies to A=b×hA = b \times h. The trapezoid formula is a more general version that works for any quadrilateral with parallel sides.

Glossary

Trapezoid
A quadrilateral with exactly one pair of parallel sides
Bases
The two parallel sides of a trapezoid, labeled b1b_1 and b2b_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=(b1+b2)×h2A = \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.

Today's challengeOne problem, three tries

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