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Trapezoid

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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 c=d. A right trapezoid is a trapezoid having two right angles.

The area of the trapezoid is

A=1/2(a+b)h
(1)
=mh
(2)
=1/4(b+a)/(b-a)eta,
(3)

where

eta=sqrt((-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d))
(4)
=4sqrt((s-a)(s-b)(s-b-c)(s-b-d)),
(5)

and

s=1/2(a+b+c+d)
(6)

is the semiperimeter.

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

x^_=b/2+((2a+b)(c^2-d^2))/(6(b^2-a^2))
(7)
y^_=(b+2a)/(3(a+b))h
(8)
=(b+2a)/(6(b^2-a^2))eta
(9)

(cf. Harris and Stocker 1998, p. 83, who give y^_ but not x^_).

The trapezoid depicted has central median

m=1/2(a+b).
(10)

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

x_1=((b-a)^2+c^2-d^2)/(2(b-a))
(11)
x_2=((b-a)^2-c^2+d^2)/(2(b-a))
(12)

respectively. This gives the vertex angles as

theta_1=cos^(-1)(((b-a)^2+c^2-d^2)/(2(b-a)c))
(13)
theta_2=cos^(-1)(((b-a)^2-c^2+d^2)/(2(b-a)c))
(14)
theta_3=cos^(-1)(((a-b)^2-c^2+d^2)/(2(a-b)c))
(15)
theta_4=cos^(-1)(((a-b)^2+c^2-d^2)/(2(a-b)c))
(16)

from the lower left corner proceeding counterclockwise.

TrapezoidDiagonals

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

p=sqrt((ab^2-a^2b-ac^2+bd^2)/(b-a))
(17)
q=sqrt((ab^2-a^2b-ad^2+bc^2)/(b-a))
(18)

and the height by

h=eta/(2|b-a|),
(19)

with eta as defined above.

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

x_0=(b(a^2-b^2-c^2+d^2))/(2(a^2-b^2))
(20)
y_0=(beta)/(2(b^2-a^2)),
(21)

The areas of the indicated triangles are

Delta_a=(a^2)/(2(a+b))h
(22)
Delta_b=(b^2)/(2(a+b))h
(23)
Delta_c=(ab)/(2(a+b))h
(24)
Delta_d=(ab)/(2(a+b))h,
(25)

so

Delta_c=Delta_d
(26)

and

Delta_aDelta_b=Delta_cDelta_d=(a^2b^2)/(4(a+b)^2)h^2
(27)

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

See also

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.

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Trapezoid

Cite this as:

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

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