Brahmagupta's formula

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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.

Formula[edit]

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

Proof[edit]

Diagram for reference

Trigonometric proof[edit]

Here the notations in the figure to the right are used. The area K of the cyclic quadrilateral equals the sum of the areas of ADB and BDC:

But since ABCD is a cyclic quadrilateral, DAB = 180° − ∠DCB. Hence sin A = sin C. Therefore,

Solving for common side DB, in ADB and BDC, the law of cosines gives

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

Substituting this in the equation for the area,

The right-hand side is of the form a2b2 = (ab)(a + b) and hence can be written as

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

Introducing the semiperimeter S = p + q + r + s/2,

Taking the square root, we get

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 any two opposite angles. (The choice of which pair of opposite angles is irrelevant: if the other two angles are taken, half their sum is 180° − θ. Since cos(180° − θ) = −cos θ, we have cos2(180° − θ) = cos2 θ.) 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 is 90°, whence the term

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]

  • Heron's formula for the area of a triangle is the special case obtained by taking d = 0.
  • The relationship between the general and extended form of Brahmagupta's formula is similar to how the law of cosines extends the Pythagorean theorem.
  • Increasingly complicated closed-form formulas exist for the area of general polygons on circles, as described by Maley et al.[3]

References[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.
  3. ^ Maley, F. Miller; Robbins, David P.; Roskies, Julie (2005). "On the areas of cyclic and semicyclic polygons" (PDF). Advances in Applied Mathematics 34 (4): 669–689. doi:10.1016/j.aam.2004.09.008. 

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.