Trapezoid
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In geometry, a four-sided 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 
ABCD.
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[edit] Area
The area of a trapezoid is,
where h is the height, and a and b are the two base lengths. It is essentially the mid-segment of the trapezoid, 
, times the height of the trapezoid. This formula works for any trapezoid.
[edit] 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,[1][2] but this does not reflect current usage (the Oxford English Dictionary says “Often called by English writers in the 19th century”).[3]
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.[3]
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[4] define a trapezoid as a quadrilateral having exactly one pair of parallel sides, thereby excluding parallelograms. Other authors[5] define a trapezoid as a quadrilateral with at least one pair of parallel sides, making a parallelogram a special type of trapezoid.
[edit] Characteristics and properties
In an isosceles trapezoid, the base angles have the same measure, and the other pair of opposite sides AD and BC also have the same length.
A quadrilateral is a trapezoid if and only if it only has 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, as long as that ratio is different from 1; 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:
When the smaller parallel side c is set to zero, this formula reduces to Heron's formula.
The line joining the mid-points of the parallel sides bisects the area.
If the trapezoid above is divided into 4 triangles by its diagonals AC and BD, intersecting at O, then the area of ΔAOD is equal to that of ΔBOC, and the product of the areas of ΔAOD and ΔBOC is equal to that of ΔAOB and ΔCOD. The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides.
[edit] 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.
[edit] See also
[edit] References
- ^ Chambers 21st Century Dictionary Trapezoid
- ^ "1913 American definition of trapezium". Merriam-Webster Online Dictionary. http://www.merriam-webster.com/dictionary/trapezium. Retrieved on 2007-12-10.
- ^ a b Oxford English Dictionary entries for trapezoid and trapezium.
- ^ "American School definition from "math.com"". http://www.math.com/school/glossary/defs/trapezoid.html. Retrieved on 2008-04-14.
- ^ ""Trapezoid" on MathWorld". http://mathworld.wolfram.com/Trapezoid.html. Retrieved on 2008-04-14.
[edit] 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]
