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# Trapezoid

A trapezoid is a quadrilateral with two sides parallel. The trapezoid is equivalent to the British definition of trapezium (Bronshtein and Semendyayev 1977, p. 174). An isosceles trapezoid is a trapezoid in which the base angles are equal so . A right trapezoid is a trapezoid having two right angles.

The area of the trapezoid is

 (1) (2) (3)

The geometric centroid lies on the median between the base and top, and if the lower left-hand corner of the trapezoid is at the original, lies at

 (4) (5) (6)

(cf. Harris and Stocker 1998, p. 83, who give but not ).

The trapezoid depicted has central median

 (7)

If vertical lines are extended from the endpoints of the upper side, the bases of the triangles formed on the left and right are

 (8) (9)

respectively. This gives the vertex angles as

 (10) (11) (12) (13)

from the lower left corner proceeding counterclockwise.

In terms of the side length, the diagonals of the trapezoid are given by

 (14) (15)

and the height by

 (16)

where

 (17) (18)

where

 (19)

is the semiperimeter.

Letting the lower left vertex be located at the origin, the intersection of the diagonals occurs at

 (20) (21)

The areas of the indicated triangles are

 (22) (23) (24) (25)

so

 (26)

and

 (27)

(B. Gladman, pers. comm., Apr. 20, 2006).

Isosceles Trapezoid, Parallelogram, Pyramidal Frustum, Rectangle, Right Trapezoid, Strombus, Trapezium Explore this topic in the MathWorld classroom

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## References

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 123, 1987.Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 3rd ed. New York: Springer-Verlag, 1997.Harris, J. W. and Stocker, H. "Trapezoid." §3.6.2 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 82-83, 1998.Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 3, 1948.

## Cite this as:

Weisstein, Eric W. "Trapezoid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Trapezoid.html