A trapezoid is a quadrilateral with exactly one pair of parallel sides, called the bases.
Quick Reference:
Definition: A four-sided polygon with exactly one pair of parallel sides.
Bases: The two parallel sides ($b_1$ and $b_2$)
Legs: The two non-parallel sides
Height ($h$): The perpendicular distance between the two bases
Area formula: $A = \dfrac{1}{2}(b_1 + b_2) \times h$
Perimeter: $P = b_1 + b_2 + l_1 + l_2$ (sum of all four sides)
Midsegment: $m = \dfrac{b_1 + b_2}{2}$ (parallel to both bases)
Type: Quadrilateral — plane geometry
Note: Called "trapezium" in the United Kingdom, Australia, and most of Europe
Full Definition
A trapezoid is a quadrilateral in which one pair of opposite sides is parallel. The parallel sides are the bases; the non-parallel sides are the legs (also called lateral sides). The perpendicular height $h$ is measured between the two bases — not along a leg.
In the United States and Canada, the shape is called a trapezoid. In the United Kingdom, Australia, and most non-American curricula, the same shape is called a trapezium. The naming difference results from a historical mix-up in the 18th century.
Types of Trapezoids
Type | Description |
|---|---|
Isosceles trapezoid | Legs are equal; base angles are equal; diagonals are equal |
Right trapezoid | One leg is perpendicular to both bases (one right angle) |
Scalene trapezoid | No additional symmetry — legs and angles all different |
Key Properties
Every trapezoid has exactly one pair of parallel sides. The co-interior angles (same-side interior angles) between the bases and one leg always add up to 180°. The midsegment — the segment connecting the midpoints of the two legs — is parallel to both bases and has length equal to the average: $m = \frac{b_1 + b_2}{2}$.
The area formula $A = \frac{1}{2}(b_1 + b_2) \times h$ can be understood as: the trapezoid is equivalent in area to a rectangle whose width equals the midsegment (the average of the two bases) and whose height equals $h$.
Origin
The word "trapezoid" derives from the Greek trapeza (τράπεζα, "table") — the shape resembles a table when the longer base is at the bottom. Euclid (c. 300 BCE) classified the trapezoid in Elements among the quadrilaterals. Babylonian astronomers used trapezoidal area calculations around 1800 BCE to track planetary motion — a discovery documented in a 2016 Science paper by historian Mathieu Ossendrijver.
Worked Examples of Trapezoid
Example 1: Area of a trapezoid
A trapezoid has parallel sides of 8 cm and 14 cm, and a height of 6 cm.
$$A = \frac{1}{2}(b_1 + b_2) \times h = \frac{1}{2}(8 + 14) \times 6 = \frac{1}{2} \times 22 \times 6 = 66 \text{ cm}^2$$
Final answer: Area = 66 cm²
Example 2: Perimeter of a trapezoid
A trapezoid has parallel sides 5 cm and 11 cm, and legs 6 cm and 8 cm.
$$P = 5 + 11 + 6 + 8 = 30 \text{ cm}$$
Final answer: Perimeter = 30 cm
Common Confusions: What is a Trapezoid vs Related Shapes
Trapezoid vs trapezium is a regional naming difference, not a mathematical one. The shape with exactly one pair of parallel sides is called a trapezoid in the US and a trapezium in the UK. Check your curriculum's terminology before answering exam questions.
The height $h$ is not the leg length. The height is the perpendicular distance between the two bases, which equals the leg length only in a right trapezoid (where one leg is perpendicular to the bases).
A parallelogram is not a trapezoid under the exclusive definition — a parallelogram has two pairs of parallel sides; a trapezoid has exactly one. Under the inclusive definition used in some curricula, a parallelogram is a special case of a trapezoid.
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