Trapezoid

In geometry, a figure with one pair of parallel sides is referred to as trapezoid in American English, and as a trapezium in British English. A trapezoid with vertices ABCD is denoted.

Definition and terminology
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 sense is the one that is standard in the U.S., but in practice quadrilateral is used rather than trapezium.

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.

Characteristics and properties
In an isosceles trapezoid, the base angles are equal, and so are the other pair of opposite sides AD and BC.

If sides AD and BC are also parallel, then they are equal, and the trapezoid is also a parallelogram. Otherwise, the other two opposite sides may be extended until they meet at a point, forming a triangle containing the trapezoid.

A quadrilateral is a trapezoid if and only if two adjacent angles that are supplementary, that is, they add up to one straight angle of 180 degrees (π radians). Another necessary and sufficient condition is that the diagonals cut each other in mutually the same ratio; this ratio is the same as that between the lengths of the parallel sides.

The mid-segment (occasionally referred to as the median) of a trapezoid is the segment that joins the midpoints of the other pair of opposite sides. The area of the trapezium is equal to the length of this mid-segment multiplied by the perpendicular height. Another formula for the area can be used when all that is known are the lengths of the four sides. If the sides are a, b, c and d, and a and c are parallel (where a is the longer parallel side), then:


 * $$A=\frac{a+c}{4(a-c)}\sqrt{(a+b-c+d)(a-b-c+d)(a+b-c-d)(-a+b+c+d)}$$


 * $$A= \frac{a+c}{4(a-c)}\sqrt{4a^3c-a^4+2a^2b^2-6a^2c^2+2a^2d^2-4ab^2c+4ac^3-4acd^2-b^4+2b^2c^2+2b^2d^2-c^4+2c^2d^2-d^4}$$

This formula does not work when the parallel sides a and c are equal since we would have division by zero. In this case the trapezoid is necessarily a parallelogram (and so $$b = d$$) and the numerator of the formula would also equal zero. In fact, the sides of a parallelogram aren't enough to determine its shape or area, the area of a parallelogram with side lengths a and b can be any number from $$ab$$ to 0.

When the smaller parallel side c is set to 0, this formula reduces to Heron's formula.

The line joining the mid-points of the parallel sides (which could also be called the median) bisects the area.

A better way to find the area of a trapezoid is:


 * $$A= \frac{h(b_1 + b_2)}{2}$$

If the trapezoid above is divided into 4 triangles by its diagonals AC and BD, intersecting at O, then the area of &Delta;AOD is equal to that of &Delta;BOC, and the product of the areas of &Delta;AOD and &Delta;BOC is equal to that of &Delta;AOB and &Delta;COD. The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides.



Architecture
In architecture the word is used to refer to symmetrical doors, windows, and buildings built wider at the base, tapering towards the top, in Egyptian style.