Trapezoid


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For other uses, see Trapezoid (disambiguation) and Trapezium (disambiguation).
A trapezoid

A trapezoid (in North America) or a trapezium (in Britain and elsewhere) is a quadrilateral (a closed plane shape with four linear sides) that has at least one pair of parallel lines for sides.

Contents

Terminology

Some authors[1] define it as a quadrilateral having exactly one pair of parallel sides, so as to exclude parallelograms, which otherwise would be regarded as a special type of trapezoid, but most mathematicians use the inclusive definition.[2]

In North America, the term trapezium is used to refer to a quadrilateral with no parallel sides. The term trapezoid is sometimes still defined as a quadrilateral without any parallel sides in Britain and elsewhere[3][4], but the current use of the word in English is usually in the American sense[citation needed]. A trapezoid with vertices ABCD would be denoted as ABCD.

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 only 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.A, but in practice quadrilateral is used rather than trapezium.[5]

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 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 it contains 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. It is parallel to the two parallel sides, and its length is the arithmetic mean of the lengths of those sides. The line joining the mid-points of the parallel sides (which could also be called the median) bisects the area.

Trapezoid, with height (h) and parallel sides (a,b) marked. The area is equal to A= h(\frac{a + b}{2}).

The area of a trapezoid can be computed as the length of the mid-segment, multiplied by the distance along a perpendicular line between 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.

Thus, if a and b are the lengths of the two parallel sides and h is the distance (height) between the parallels, the area formula is as follows:

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

The quantity \frac{a + b}{2} is the average of the horizontal lengths of the trapezoid, so the area can be understood to be the product of the height and average length of the shape.

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)}.

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 zero, this formula reduces to Heron's formula.

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.

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.

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

References

  1. ^ "American School definition from "math.com"". Retrieved on 2008-04-14.
  2. ^ ""Trapezoid" on MathWorld". Retrieved on 2008-04-14.
  3. ^ Chambers 21st Century Dictionary Trapezoid
  4. ^ "1913 American definition of trapezium". Merriam-Webster Online Dictionary. Retrieved on 2007-12-10.
  5. ^ Oxford English Dictionary entries for trapezoid and trapezium.

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