Trapezium - Definition, Properties, Area and Examples

Trapezium - Definition, Properties, Area and Examples

What Is a Trapezium?

In UK / Indian usage: a trapezium is a quadrilateral with exactly one pair of parallel sides.
In US usage, the same shape is called a trapezoid — and the word trapezium means no parallel sides.
For clarity, this article uses UK terminology.
The parallel sides are the bases (usually a and b). The non-parallel sides are the legs. The perpendicular distance between the bases is the height h.

Types of Trapezium

Scalene Trapezium

All four sides have different lengths. The most general type.

Isosceles Trapezium

The two non-parallel sides (legs) are equal in length. Has an axis of symmetry perpendicular to the bases. Both base angles at each base are equal.

Right Trapezium

Has two right angles (one of the legs is perpendicular to the bases).

Area Formula

A = \tfrac{1}{2}(a + b) h
where a and b are the lengths of the two parallel sides (bases) and h is the perpendicular height between them.
Interpretation: The area is the average of the two bases multiplied by the height.

Perimeter Formula

P = a + b + c + d
where a and b are the bases and c and d are the legs. Just add all four side lengths.

Three Worked Examples — Quick, Standard, Stretch

Quick — Area

A trapezium has parallel sides 8 cm and 12 cm, and height 5 cm. Find its area. A = \tfrac{1}{2}(8 + 12)(5) = \tfrac{1}{2}(20)(5) = 50 cm²

Standard — Find Missing Base

A trapezium has area 40 m², one base 6 m, and height 4 m. Find the other base. 40 = \tfrac{1}{2}(6 + b)(4) = 2(6 + b)
20 = 6 + b
b = 14 m.

Stretch — Isosceles Trapezium Properties

An isosceles trapezium has bases 10 cm and 4 cm, with legs of length 5 cm. Find its height and area.
The horizontal "overhang" on each side is (10−4)/2 = 3 cm.
Using the Pythagorean theorem:

h^2 + 3^2 = 5^2 \implies h^2 = 16 \implies h = 4 cm
Area: A = \tfrac{1}{2}(10 + 4)(4) = \tfrac{1}{2}(14)(4) = 28 cm².

Properties of an Isosceles Trapezium

Why Does the Trapezium Matter?

Trapeziums appear in:

A Worked Example

Find the area of a trapezium with bases 7 cm and 11 cm, and slant height (leg length) 5 cm. The intuitive (wrong) approach: A student uses the slant height 5 as the height.

Why it fails: The "height" in the area formula is the perpendicular distance between the parallel sides — not the slant length of a leg.

What Are the Most Common Mistakes With the Trapezium?

Mistake 1: Using the leg length as the height

The fix: Height = perpendicular distance between the parallel sides.

Mistake 2: Forgetting the factor of \tfrac{1}{2}

The fix: Area is \tfrac{1}{2}(a + b)h.

Mistake 3: Adding both bases without averaging

The fix: Use the average of the bases times the height.

Key Takeaways