For a general quadrilateral with sides of length 
,
,
,
and 
,
the area 
is given by
![K=sqrt((s-a)(s-b)(s-c)(s-d)-abcdcos^2[1/2(A+B)]),](/images/equations/BrahmaguptasFormula/NumberedEquation1.svg) |
(1)
|
where
 |
(2)
|
is the semiperimeter, 
is the angle between 
and 
, and 
is the angle between 
and 
. Brahmagupta's formula
 |
(3)
|
is a special case giving the area of a cyclic quadrilateral (i.e., a quadrilateral inscribed
in a circle), for which 
. In terms of the circumradius 
of a cyclic
quadrilateral,
 |
(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
(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.MathPages. "Heron's
Formula and Brahmagupta's Generalization." http://www.mathpages.com/home/kmath196.htm.Referenced
on Wolfram|Alpha
Brahmagupta's Formula
Cite this as:
Weisstein, Eric W. "Brahmagupta's Formula."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BrahmaguptasFormula.html
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