Trapezoid

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A 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 Template:Trapezoidnotation.

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.

Area

The area of a trapezoid is given by

$A={\frac {h(a+b)}{2}}$

where $h$ is the height, and $a, b$ are the lengths of the parallel sides. This yields as a special case the well-known formula for the area of a triangle, by considering a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point.

The mid-segment of a trapezoid is the segment that joins the midpoints of the non-parallel sides. Its length $m$ is equal to the average of the lengths of the bases of the trapezoid:

$m={\frac {a+b}{2}}$

It follows that the area of a trapezoid is equal to the length of this mid-segment multiplied by the height:

$A=mh$

In the case that the two parallel sides are different lengths ($a \ne b$), the height of a trapezoid $h$ , and hence its area $A$ can be determined by the length of all of its sides:

$h={\frac {\sqrt {(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}}{2(b-a)}}$

where, $c,d$ are the lengths of the other two sides, and hence

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

When one of the parallel sides has shrunk to a point (say $a = 0$), this formula reduces to Heron's formula for the area of a triangle.

Another equivalent formula for the area, which more closely resembles Heron's formula is:

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

where

$s = \frac{a+b+c+d}{2}$

is the semiperimeter of the trapezoid. (This formula is similar to Brahmagupta's formula, but it differs from it, in that a trapezoid might not be cyclic (inscribed in a circle). The formula is also a special case of Bretschneider's formula for a general quadrilateral).

Therefore using Bretschneider's formula gives:

$A={\sqrt {{\dfrac {(ab^{2}-a^{2}b-ad^{2}+bc^{2})(ab^{2}-a^{2}b-ac^{2}+bd^{2})}{4(b-a)^{2}}}-{\frac {(b^{2}+d^{2}-a^{2}-c^{2})^{2}}{16}}}}$

One can write expanded formulas for the area, but they are harder to remember because they have less symmetry.

Diangle lengths

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

Therefore the diangle lengths are:

$p={\sqrt {\frac {ab^{2}-a^{2}b-ac^{2}+bd^{2}}{b-a}}}$

$q={\sqrt {\frac {ab^{2}-a^{2}b-ad^{2}+bc^{2}}{b-a}}}$

The Temple of Dendur in the Metropolitan Museum of Art, New York

Circumcircle of a Trapezoid

The circumradius of a trapezoid is

$R={\frac {a-c}{a+c}}{\sqrt {\frac {(ac+bd){\bigl (}ac(b^{2}+d^{2})+bd(a^{2}+c^{2}){\bigr )}}{(P-2c)(P-2a)(P-2c-2b)(P-2c-2d)}}}$

Incircle of a Trapezoid

The inradius of a trapezoid is

$r={\dfrac {a+c}{2}}{\sqrt {\frac {(P-2a)(P-2c)(P-2c-2d)(P-2c-2b)}{(a^{2}+(a-c)(b+d)-c^{2})^{2}}}}$

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.

See also

• Trapezoidal number
• Isosceles trapezoid

External links

• "Trapezoid" on MathWorld
• Trapezoid definition   Area of a trapezoid   Median of a trapezoid With interactive animations
• Trapezoid (North America) at elsy.at: Animated course (construction, circumference, area)
• [1] on Numerical Methods for Stem Undergraduate
• Autar Kaw and E. Eric Kalu, Numerical Methods with Applications, (2008) [2]

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