Area Of A Trapezoid Calculator

What Is a Trapezoid?

A trapezoid (called a trapezium in British English) is a quadrilateral with exactly one pair of parallel sides. The two parallel sides are the bases, denoted b₁ and b₂, while the two non-parallel sides are the legs. The perpendicular distance between the bases is the height (h). Trapezoids appear frequently in architecture, land surveying, and civil engineering, making their area calculation a widely practical skill.

The Area of a Trapezoid Formula

The standard formula for the area of a trapezoid is:

A = ½ × (b₁ + b₂) × h

This formula averages the two parallel bases and multiplies by the perpendicular height. The variables are defined as follows:

• A — Area of the trapezoid, expressed in squared length units (e.g., cm², ft², m²)
• b₁ — Length of the first parallel base
• b₂ — Length of the second parallel base
• h — Perpendicular height: the shortest distance between the two bases, measured at a right angle to both

Deriving the Formula

Two geometric proofs establish the trapezoid area formula. The first duplicates the trapezoid, rotates the copy 180°, and attaches it alongside the original to form a parallelogram with base (b₁ + b₂) and height h. Because a parallelogram's area equals base × height, and the original trapezoid is exactly half the parallelogram, the trapezoid area equals ½ × (b₁ + b₂) × h. This intuitive proof is demonstrated clearly by Khan Academy's trapezoid area video.

The second proof draws a diagonal across the trapezoid, splitting it into two triangles. The first triangle has base b₁ and height h, yielding area ½b₁h. The second has base b₂ and height h, yielding area ½b₂h. Adding both confirms: A = ½b₁h + ½b₂h = ½(b₁ + b₂)h. Full derivation details appear in the Area of Trapezoid Formula reference (PDF).

Worked Example

Consider a trapezoid with the following measurements:

• Base 1 (b₁) = 14 meters
• Base 2 (b₂) = 8 meters
• Height (h) = 6 meters

Applying the formula step by step:

• Add the bases: 14 + 8 = 22 m
• Multiply by height: 22 × 6 = 132 m²
• Divide by 2: 132 ÷ 2 = 66 m²

The area of this trapezoid is 66 square meters. Converting to feet (b₁ ≈ 45.9 ft, b₂ ≈ 26.2 ft, h ≈ 19.7 ft), the same calculation yields approximately 710.5 square feet, illustrating how unit choice changes the numerical result without altering the physical area.

Real-World Applications

The trapezoid area formula has widespread practical uses across many fields:

• Construction and architecture: Roof cross-sections, gabled walls, and trapezoidal floor tiles all require accurate area calculations for material ordering and cost estimation.
• Land surveying: Irregular plots bounded by roads or rivers often approximate trapezoids. Surveyors apply this formula to estimate land area for taxation and property sales.
• Civil engineering: Drainage ditches, highway embankments, and canal cross-sections are frequently trapezoidal. Engineers calculate these cross-sectional areas to determine flow capacity and earthwork volumes.
• Numerical integration: The trapezoid rule in calculus applies the same averaging logic to approximate definite integrals, treating curve segments as trapezoids whose combined areas estimate the integral's value.

Special Cases

The trapezoid formula generalizes several simpler geometric shapes:

• When b₁ = b₂: The shape becomes a parallelogram, and the formula reduces to A = b × h.
• When b₂ = 0: One base disappears and the shape becomes a triangle, giving A = ½ × b₁ × h.
• An isosceles trapezoid has legs of equal length and a vertical line of symmetry, commonly seen in bridge trusses and arched doorway designs.

Unit System and Accuracy

Area is always expressed in squared units. All three inputs — b₁, b₂, and h — must share the same unit system before applying the formula. Mixing units (for example, bases in meters but height in centimeters) will produce an incorrect result. This area of a trapezoid calculator automatically displays the output in the correct squared unit based on the selected unit system, removing that source of error.