Trapezoid

In geometry, a trapezoid (/ˈtræpəzɔɪd/) in North American English, or trapezium (/trəˈpiːziəm/) in British English, is a quadrilateral that has at least one pair of parallel sides.

The parallel sides are called the bases of the trapezoid. The other two sides are called the legs or lateral sides. If the trapezoid is a parallelogram, then the choice of bases and legs is arbitrary.

A trapezoid is usually considered to be a convex quadrilateral in Euclidean geometry, but there are also crossed cases. If shape ABCD is a convex trapezoid, then ABDC is a crossed trapezoid. The metric formulas in this article apply in convex trapezoids.

Trapezoid can be defined exclusively or inclusively. Under an exclusive definition a trapezoid is a quadrilateral having exactly one pair of parallel sides, with the other pair of opposite sides non-parallel. Parallelograms including rhombi, rectangles, and squares are then not considered to be trapezoids. Under an inclusive definition, a trapezoid is any quadrilateral with at least one pair of parallel sides. In an inclusive classification scheme, definitions are hierarchical: a square is a type of rectangle and a type of rhombus, a rectangle or rhombus is a type of parallelogram, and every parallelogram is a type of trapezoid.

Professional mathematicians and post-secondary geometry textbooks nearly always prefer inclusive definitions and classifications, because they simplify statements and proofs of geometric theorems. In primary and secondary education, definitions of rectangle and parallelogram are also nearly always inclusive, but an exclusive definition of trapezoid is commonly found. This article uses the inclusive definition and considers parallelograms to be special kinds of trapezoids. (cf. Quadrilateral § Taxonomy)

To avoid confusion, some sources use the term proper trapezoid to describe trapezoids with exactly one pair of parallel sides, analogous to uses of the word proper in some other mathematical objects.

In the ancient Greek geometry of Euclid's Elements (c. 300 BC), quadrilaterals were classified into exclusive categories: square; oblong (non-square rectangle); (non-square) rhombus; rhomboid, meaning a non-rhombus non-rectangle parallelogram; or trapezium (τραπέζιον, literally "table"), meaning any quadrilateral not already included in the previous categories.

The Neoplatonist philosopher Proclus (mid 5th century AD) wrote an influential commentary on Euclid with a richer set of categories, which he attributed to Posidonius (c. 100 BC). In this scheme, a quadrilateral can be a parallelogram or a non-parallelogram. A parallelogram can itself be a square, an oblong (non-square rectangle), a rhombus, or a rhomboid (non-rhombus non-rectangle). A non-parallelogram can be a trapezium with exactly one pair of parallel sides, which can be isosceles (with equal legs) or scalene (with unequal legs); or a trapezoid (τραπεζοειδή, literally "table-like") with no parallel sides.