Area of a Trapezoid
Reference for the trapezoid area formula A = half times (a plus b) times h.
Calculate area from two parallel sides and height with step-by-step examples.
The Formula
A = ½ × (a + b) × h
This formula calculates the area of a trapezoid (also called a trapezium in some countries).
A trapezoid has exactly one pair of parallel sides.
Variables
| Symbol | Meaning |
|---|---|
| A | Area of the trapezoid |
| a | Length of one parallel side (base 1) |
| b | Length of the other parallel side (base 2) |
| h | Perpendicular height between the two parallel sides |
Example 1
Find the area of a trapezoid with parallel sides of 8 cm and 12 cm, and height 5 cm
a = 8, b = 12, h = 5
A = ½ × (8 + 12) × 5
A = ½ × 20 × 5 = ½ × 100
A = 50 cm²
Example 2
A garden bed is shaped like a trapezoid with parallel sides of 3 m and 5 m, and a height of 2.4 m
a = 3, b = 5, h = 2.4
A = ½ × (3 + 5) × 2.4
A = ½ × 8 × 2.4 = ½ × 19.2
A = 9.6 m²
When to Use It
Use the area of a trapezoid formula when:
- Calculating the area of shapes with two parallel sides of different lengths
- Working with cross-sections of channels, ditches, or embankments
- Measuring irregularly shaped land plots that have two parallel edges
- Solving geometry problems involving trapezoids
Key Notes
- Formula: A = ½(a + b)h: a and b are the two parallel sides (bases) and h is the perpendicular height between them (not the slant side length). The formula is the average of the two bases multiplied by the height.
- Special cases: When a = b, the trapezoid becomes a parallelogram (A = bh). When one base equals zero, it becomes a triangle (A = ½bh). The trapezoid formula is therefore a generalization of both.
- Midsegment theorem: The line segment connecting the midpoints of the two non-parallel sides (legs) is called the midsegment. Its length equals (a+b)/2 — exactly what appears in the area formula. The midsegment is also parallel to both bases.
- Finding the perpendicular height: If the slant side length s and base lengths are known, h can be found using the Pythagorean theorem on the right triangle formed by the slant leg, height, and horizontal offset: h = √(s² − ((b−a)/2)²) for an isosceles trapezoid.
- Applications: Trapezoidal cross-sections appear in irrigation channels, road embankments, and architectural facades. The trapezoidal rule in calculus uses this area formula to numerically approximate integrals.