English

梯形的

This article is about the geometric figure. For other uses, see Trapezoid (disambiguation).

A trapezoid (in North America) or trapezium (in Britain and elsewhere) is a quadrilateral, which is defined as a shape with four sides, which has one set of parallel sides. Some authors define it as a quadrilateral having exactly one set of parallel sides, so as to exclude parallelograms.

The exactly opposite concept, a quadrilateral that has no parallel sides, is referred to as a trapezium in North America, and as a trapezoid in Britain and elsewhere. To avoid confusion, this article uses the North American wording. It also admits parallelograms as special cases of trapezoids (however, in this case, it is assumed that one set of parallel sides is distinguished, and is the one referred to as "the set of parallel sides").

In an isosceles trapezoid, the base angles are congruent, and so are the pair of non-parallel opposite sides.

If the other set of opposite sides is 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 that the trapezoid lies inside.

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 midsegment (occasionally referred to as the median) of a trapezoid is the segment that joins the midpoints of the other set 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 area of a trapezoid can be computed as the length of the midsegment, 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 two parallel sides and h is the distance (height) between the parallels, the area formula is as follows:

A= \frac{(a+b)h}{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 average length and height 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 doesn't 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 sides a and b can be any number from "a b" to "zero".

When the smaller parallel side c is set to zero, this formula turns to be 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.