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Brahmagupta's Formula

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BrahmaguptasFormula

Brahmagupta's formula gives the area of a cyclic quadrilateral as a special case of the following general formula. For a quadrilateral with sides of length a, b, c, and d, the area K is given by

K=sqrt((s-a)(s-b)(s-c)(s-d)-abcdcos^2[1/2(A+B)]),
(1)

where

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

is the semiperimeter, A is the angle between a and d, and B is the angle between b and c. Brahmagupta's formula

K=sqrt((s-a)(s-b)(s-c)(s-d))
(3)

is a special case giving the area of a cyclic quadrilateral (i.e., a quadrilateral inscribed in a circle), for which A+B=pi. In terms of the circumradius R of a cyclic quadrilateral,

K=(sqrt((bc+ad)(ac+bd)(ab+cd)))/(4R).
(4)

The area of a cyclic quadrilateral is the maximum possible for any quadrilateral with the given side lengths.

For a bicentric quadrilateral (i.e., a quadrilateral that can be inscribed in one circle and circumscribed on another), the area formula simplifies to

K=sqrt(abcd)
(5)
=1/2sqrt(p^2q^2-(ac-bd)^2)
(6)

(Ivanoff 1960; Beyer 1987, p. 124).

See also

Bicentric Quadrilateral, Bretschneider's Formula, Cyclic Quadrilateral, Heron's Formula, Quadrilateral

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References

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 123, 1987.Coolidge, J. L. "A Historically Interesting Formula for the Area of a Quadrilateral." Amer. Math. Monthly 46, 345-347, 1939.Coxeter, H. S. M. and Greitzer, S. L. "Cyclic Quadrangles; Brahmagupta's Formula." §3.2 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 56-60, 1967.Ivanoff, V. F. "Solution to Problem E1376: Bretschneider's Formula." Amer. Math. Monthly 67, 291-292, 1960.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 81-82, 1929.Mathologer. "Heron's Formula: What Is the Hidden Meaning of 1 + 2 + 3 = 1 x 2 x 3?" May 14, 2022. https://www.youtube.com/watch?v=IguNXoCjBEk.MathPages. "Heron's Formula and Brahmagupta's Generalization." https://www.mathpages.com/home/kmath196/kmath196.htm.

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Brahmagupta's Formula

Cite this as:

Weisstein, Eric W. "Brahmagupta's Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BrahmaguptasFormula.html

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