Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, Brahmagupta's formula, named after the 7th century

Indian mathematician Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, ...

, is used to find the

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

of any convex

cyclic quadrilateral In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertex (geometry), vertices all lie on a single circle, making the sides Chord (geometry), chords of the circle. This circle is called ...

(one that can be inscribed in a circle) given the lengths of the sides. Its generalized version, ''

Bretschneider's formula In geometry, Bretschneider's formula is a mathematical expression for the area of a general quadrilateral. It works on both convex and concave quadrilaterals, whether it is cyclic or not. The formula also works on crossed quadrilaterals provided ...

'', can be used with non-cyclic quadrilateral. ''

Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths Letting be the semiperimeter of the triangle, s = \tfrac12(a + b + c), the area is A = \sqrt. It is named after first-century ...

'' can be thought as a special case of the Brahmagupta's formula for triangles.

Formulation

Brahmagupta's formula gives the area of a convex

whose sides have lengths , , , as :

K=\sqrt

where , the

semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name ...

, is defined to be :

s=\frac.

This formula generalizes

for the area of a

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as (or any one side) 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 :

K=\frac\sqrt.

Another equivalent version is :

K=\frac\cdot

Proof

Trigonometric proof

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

K = \fracpq\sin A + \fracrs\sin C.

But since is a cyclic quadrilateral, . Hence . Therefore, :

K = \fracpq\sin A + \fracrs\sin A

K^2 = \frac (pq + rs)^2 \sin^2 A

4K^2 = (pq + rs)^2 (1 - \cos^2 A) = (pq + rs)^2 - ((pq + rs)\cos A)^2

(using the

trigonometric identity In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...

). Solving for common side , in and , the

law of cosines In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides , , and , opposite respective angles , , and (see ...

gives :

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

Substituting (since angles and are supplementary) and rearranging, we have :

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

Substituting this in the equation for the area, :

4K^2 = (pq + rs)^2 - \frac(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 and hence can be written as :

(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 :

16K^2= (r+s)^2 - (p-q)^2 (p+q)^2 - (r-s)^2 ]

that can be factored again into :

16K^2=(q+r+s-p)(p+r+s-q)(p+q+s-r)(p+q+r-s).

Introducing the

yields :

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

Taking the square root, we get :

K = \sqrt.

Non-trigonometric proof

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

Extension to non-cyclic quadrilaterals

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

K=\sqrt

where 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 . Since , we have .) This more general formula is known as

. It is a property of

s (and ultimately of

inscribed angle In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. Equivalently, an ...

s) that opposite angles of a quadrilateral sum to 180°. Consequently, in the case of an inscribed quadrilateral, is 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 :

K=\sqrt

where and are the lengths of the diagonals of the quadrilateral. In a

, according to Ptolemy's theorem, and the formula of Coolidge reduces to Brahmagupta's formula.

Related theorems

for the area of a

is the special case obtained by taking . * The relationship between the general and extended form of Brahmagupta's formula is similar to how the

extends the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

. * Increasingly complicated closed-form formulas exist for the area of general polygons on circles, as described by Maley et al.

References

External links

A geometric proof
from Sam Vandervelde. * {{DEFAULTSORT:Brahmagupta's Formula Brahmagupta Theorems about quadrilaterals and circles Area