Last updated: 2026-05-18

Definition

A trapezoid is a quadrilateral with exactly one pair of parallel sides. Those two parallel sides are the bases (a and b); the perpendicular distance between them is the height (h). The same shape is called a trapezium in British English — identical geometry, different name.

The formula

For a trapezoid with parallel sides a and b and perpendicular height h, the area is:

A = (a + b) / 2 × h

The expression (a + b) / 2 is the average of the two parallel sides. Multiplying that average by the perpendicular height gives the same result as the area of a rectangle whose width is that average.

Why the average works

The midsegment — the horizontal line halfway up the trapezoid — has length exactly (a + b) / 2. A rectangle with that width and height h encloses the same area as the trapezoid: the triangular piece trimmed from the wider side fills the gap on the narrower side exactly. This is why averaging the bases, rather than using either one alone, gives the correct area.

The same relationship underlies the trapezoidal rule in integration: one step of that rule approximates the area under a curve with a single trapezoid, using this identical formula.

Trapezoid vs trapezium — terminology

The same shape goes by different names depending on the region:

Region	Name for this shape
United States, Canada	Trapezoid
United Kingdom, Australia, most Commonwealth countries	Trapezium
Europe (general)	Trapeze / Trapez / Trapèze

This calculator uses the US term trapezoid throughout, but the mathematics is identical.

Historically, British usage also applied "trapezoid" to a quadrilateral with no parallel sides (what Americans would call a trapezium), so the same word once meant the opposite shape depending on the century and country. Modern usage has largely converged on the US definition.

What counts as the height

The height h is the perpendicular (vertical) distance between the two parallel sides, measured at a right angle to both bases. It is not the length of the slanted leg.

When only the slant side s and the angle θ it makes with the base are known, the perpendicular height follows from:

h = s × sin(θ)

Substituting the slant length directly for h overstates the area, because the slant side is longer than the perpendicular distance.

Special cases

Condition	Shape becomes	Area formula simplifies to
a = b	Parallelogram	A = a × h
b = 0	Triangle	A = a / 2 × h
a = b, h = a	Square or rectangle	A = a²

The trapezoid formula is therefore the most general of the basic polygon area formulas: it contains both the parallelogram and the triangle as degenerate cases.

Worked example

A cross-section of a drainage channel is trapezoidal, with a bottom width of 3 m, a top width of 5 m, and a depth of 1.2 m.

A = (3 + 5) / 2 × 1.2 = 4 × 1.2 = 4.8 m²

A second example: a stained-glass panel has parallel edges of 8 cm and 5 cm with a perpendicular height of 4 cm.

A = (8 + 5) / 2 × 4 = 6.5 × 4 = 26 cm²

Applications

Trapezoidal cross-sections appear in civil engineering and irrigation design, where the flow capacity of an open channel depends on its cross-sectional area. The formula also covers everyday measurement of any four-sided region with one pair of parallel edges, such as a plot of land, a roof section, or a panel of glass.

Frequently Asked Questions (FAQ)

What is the formula for the area of a trapezoid?

The area of a trapezoid is A = (a + b) / 2 × h, where a and b are the two parallel sides (bases) and h is the perpendicular height between them. The expression (a + b) / 2 is the average of the two bases, and multiplying it by the height gives the area. For example, a trapezoid with a = 8 cm, b = 5 cm, and h = 4 cm has area = (8 + 5) / 2 × 4 = 6.5 × 4 = 26 cm².

What is the difference between a trapezoid and a trapezium?

The two words refer to the same shape — a quadrilateral with exactly one pair of parallel sides — but the terminology differs by region. In the United States and Canada the shape is called a trapezoid, while in the United Kingdom, Australia, and most other English-speaking countries it is called a trapezium. This calculator uses "trapezoid" as the primary term, but the formula and result are identical for either word.

Is the height the slant side or the perpendicular distance?

The height h in the formula is always the perpendicular (vertical) distance between the two parallel sides, not the length of the slanted leg. When only the slant side length is known, the angle it makes with the base gives the perpendicular height through h = slant × sin(angle). Substituting the slant side directly overstates the area, because the slant side is longer than the perpendicular distance.

Is a parallelogram a special case of a trapezoid?

Yes. When both parallel sides are equal (a = b), the trapezoid becomes a parallelogram and the formula simplifies to A = a × h, which is exactly the parallelogram area formula. Similarly, if one base is zero (b = 0), the shape becomes a triangle and the formula simplifies to A = a / 2 × h, which matches the triangle area formula. The trapezoid formula is therefore a general case that covers both shapes.