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

Contents

FormulaEdit

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

 

ProofEdit

 
Diagram for reference

Trigonometric proofEdit

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 proofEdit

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

Extension to non-cyclic quadrilateralsEdit

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 theoremsEdit

  • 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]

ReferencesEdit

  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 linksEdit