trapezium

A trapezium is a four-sided polygon (a quadrilateral). The exact meaning of the term varies by regional usage. In American English, a trapezoid is defined as a quadrilateral with at least one pair of parallel sides, and a trapezium is a quadrilateral with no parallel sides. In British English, the terms are used inconsistently, and some sources use trapezium for the shape American authors call a trapezoid. Because of this variation, it is important to check the convention in a given text.

When a trapezium is defined as having one pair of parallel sides (the American notion of a trapezoid), the parallel sides are called the bases and the distance between them is the height. The area is A = (a + b)/2 × h, where a and b are the lengths of the two bases and h is the perpendicular distance between them. If the nonparallel sides are equal in length, the figure is an isosceles trapezoid and its diagonals are equal in length. If one leg is perpendicular to the bases, the figure is a right trapezoid.

If the term is used for a quadrilateral with no parallel sides (the British use historically associated with trapezium), there is no simple general formula for area; the area must be computed by decomposing the shape into triangles or using other techniques.

The word trapezium also appears in numerical analysis as the trapezium (or trapezoidal) rule, a method for approximating definite integrals by replacing the area under a curve with a series of trapezoids.

Etymology traces to Latin trapezium, from Greek trapezeion, meaning a small table.