Trapezoid for Junior High School

In geometry, a four-sided figure with one pair of parallel sides is referred to as a trapezoid in American English and as a trapezium in English outside North America. A trapezoid with vertices ABCD is denoted ABCD.

This article uses the term trapezoid in the sense that is current in the United States (and sometimes in some other English-speaking countries). Readers in the United Kingdom and Australia should read trapezium for each use of trapezoid in the following paragraphs. In all other languages using a word derived from the Greek for this figure, the form closest to trapezium (e.g. French 'trapèze', Italian 'trapezio', German 'Trapez', Russian 'трапеция') is used.

The term trapezium has been in use in English since 1570, from Late Latin trapezium, from Greek trapezion, literally "a little table", diminutive of trapeza "table", itself from tra- "four" + peza "foot, edge". The first recorded use of the Greek word translated trapezoid (τραπεζοειδη, table-like) was by Marinus Proclus (412 to 485 AD) in his Commentary on the first book of Euclid’s Elements.

There is also some disagreement on the allowed number of parallel sides in a trapezoid. At issue is whether parallelograms, which have two pairs of parallel sides, should be counted as trapezoids. Some authors define a trapezoid as a quadrilateral having exactly one pair of parallel sides, thereby excluding parallelograms. Other authors define a trapezoid as a quadrilateral with at least one pair of parallel sides, making the parallelogram a special type of trapezoid (along with the rhombus, the rectangle and the square). The latter definition is consistent with its uses in higher mathematics such as calculus. The former definition would make such concepts as the trapezoidal approximation to a definite integral be ill-defined.

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In North America, the term trapezium is used to refer to a quadrilateral with no parallel sides. The term trapezoid has been defined as a quadrilateral without any parallel sides in Britain and elsewhere, but this does not reflect current usage (the Oxford English Dictionary says “Often called by English writers in the 19th century”).
According to the Oxford English Dictionary, the trapezoid as a figure with no sides parallel is the sense for which Proclus introduced the term; it is retained in the French "trapézoïde", German "trapezoïd", and in other languages. A trapezium in Proclus' sense is a quadrilateral having one pair of its opposite sides parallel. This was the specific sense in England in 17th and 18th centuries, and again the prevalent one in recent use. A trapezium as any quadrilateral more general than a parallelogram is the sense of the term in Euclid. The sense of a trapezium as an irregular quadrilateral having no sides parallel was the usual sense in England from c1800 to c1875, but is now rare. This article uses the term trapezoid in the sense that is current in the USA and some other English-speaking countries. Readers in the UK should read trapezium for each use of trapezoid in the following paragraphs.
There is also some disagreement on the allowed number of parallel sides in a trapezoid. At issue is whether parallelograms, which have two pairs of parallel sides, should be counted as trapezoids. Some authors define a trapezoid as a quadrilateral having exactly one pair of parallel sides, thereby excluding parallelograms. Other authors define a trapezoid as a quadrilateral with at least one pair of parallel sides, making a parallelogram a special type of trapezoid.