trapeziums

A trapezium is a quadrilateral that has one pair of parallel sides, called the bases. The other two sides are the legs. The distance between the bases is called the height. In British and many European geometries, this is the standard meaning of trapezium; in American usage the terminology can vary, and some sources use trapezoid for the same shape.

Area is often the main quantity associated with a trapezium. The area is given by A = (a + b) / 2 × h, where a and b are the lengths of the two bases and h is the perpendicular distance between them. Height is the key measure, and it can be found by dropping perpendiculars from the endpoints of one base to the other base, or by using coordinate geometry. A common construction in problems is to decompose the trapezium into a rectangle and a triangle.

Properties and variants include that the base angles on the same base are supplementary (they sum to 180 degrees). If the non-parallel sides (the legs) are equal, the trapezium is isosceles, its base angles are equal, and its diagonals are usually equal as well. A right trapezium has a leg perpendicular to the bases, yielding two right angles at adjacent vertices. In general, diagonals are not equal in a trapezium, except in the isosceles case.

Trapeziums relate to other quadrilaterals through terminology that varies by region; some texts treat trapezium and trapezoid as synonyms, while others distinguish them. See also trapezoid.