Brahmagupta's formula

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In Euclidean geometry, Brahmagupta's formula finds the area of any cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides.


Brahmagupta's formula gives the area K of a cyclic quadrilateral whose sides have lengths a, b, c, d as


where s, the semiperimeter, is defined to be


This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula.

If the semiperimeter is not used, Brahmagupta's formula is


Another equivalent version is



Diagram for reference

Trigonometric proof[edit]

Here the notations in the figure to the right are used. Area K of the cyclic quadrilateral = Area of \triangle ADB + Area of \triangle BDC

= \frac{1}{2}pq\sin A + \frac{1}{2}rs\sin C.

But since ABCD is a cyclic quadrilateral, \angle DAB = 180^\circ - \angle DCB. Hence \sin A = \sin C. Therefore,

K = \frac{1}{2}pq\sin A + \frac{1}{2}rs\sin A
K^2 = \frac{1}{4} (pq + rs)^2 \sin^2 A
4K^2 = (pq + rs)^2 (1 - \cos^2 A) = (pq + rs)^2 - (pq + rs)^2 \cos^2 A.\,

Solving for common side DB, in \triangleADB and \triangle BDC, the law of cosines gives

p^2 + q^2 - 2pq\cos A = r^2 + s^2 - 2rs\cos C. \,

Substituting \cos C = -\cos A (since angles A and C are supplementary) and rearranging, we have

2 (pq + rs) \cos A = p^2 + q^2 - r^2 - s^2. \,

Substituting this in the equation for the area,

4K^2 = (pq + rs)^2 - \frac{1}{4}(p^2 + q^2 - r^2 - s^2)^2
16K^2 = 4(pq + rs)^2 - (p^2 + q^2 - r^2 - s^2)^2.

The right-hand side is of the form a^2-b^2 = (a-b)(a+b) and hence can be written as

[2(pq + rs) - p^2 - q^2 + r^2 +s^2][2(pq + rs) + p^2 + q^2 -r^2 - s^2] \,

which, upon rearranging the terms in the square brackets, yields

= [ (r+s)^2 - (p-q)^2 ][ (p+q)^2 - (r-s)^2 ] \,
= (q+r+s-p)(p+r+s-q)(p+q+s-r)(p+q+r-s). \,

Introducing the semiperimeter S = \frac{p+q+r+s}{2},

16K^2 = 16(S-p)(S-q)(S-r)(S-s). \,

Taking the square root, we get

K = \sqrt{(S-p)(S-q)(S-r)(S-s)}.

Non-trigonometric proof[edit]

An alternative, non-trigonometric proof utilizes two applications of Heron's triangle area formula on similar triangles.[1]

Extension to non-cyclic quadrilaterals[edit]

In the case of non-cyclic quadrilaterals, Brahmagupta's formula can be extended by considering the measures of two opposite angles of the quadrilateral:


where θ is half the sum of two opposite angles. (The choice of which pair of opposite angles is irrelevant: if the other two angles are taken, half their sum is the supplement of θ. Since cos(180° − θ) = −cosθ, we have cos2(180° − θ) = cos2θ.) This more general formula is known as Bretschneider's formula.

It is a property of cyclic quadrilaterals (and ultimately of inscribed angles) that opposite angles of a quadrilateral sum to 180°. Consequently, in the case of an inscribed quadrilateral, θ = 90°, whence the term

abcd\cos^2\theta=abcd\cos^2 \left(90^\circ\right)=abcd\cdot0=0, \,

giving the basic form of Brahmagupta's formula. It follows from the latter equation that the area of a cyclic quadrilateral is the maximum possible area for any quadrilateral with the given side lengths.

A related formula, which was proved by Coolidge, also gives the area of a general convex quadrilateral. It is[2]


where p and q are the lengths of the diagonals of the quadrilateral. In a cyclic quadrilateral, pq=ac+bd according to Ptolemy's theorem, and the formula of Coolidge reduces to Brahmagupta's formula.

Related theorems[edit]


  1. ^ Hess, Albrecht, "A highway from Heron to Brahmagupta", Forum Geometricorum 12 (2012), 191–192.
  2. ^ J. L. Coolidge, "A Historically Interesting Formula for the Area of a Quadrilateral", American Mathematical Monthly, 46 (1939) pp. 345-347.

External links[edit]

This article incorporates material from proof of Brahmagupta's formula on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.