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In
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
,
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the '' Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tr ...
's formula is used to find the
area Area is the quantity that expresses the extent of a region on the plane or on a curved 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 op ...
of any
cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...
(one that can be inscribed in a circle) given the lengths of the sides; its generalized version ( Bretschneider's formula) 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 , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century ...
can be thought as a sub-case of the Brahmagupta's formula.


Formula

Brahmagupta's formula gives the area of a
cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...
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 ...
, is defined to be : s=\frac. This formula generalizes
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 , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century ...
for the area of a
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as 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 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) (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, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
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 :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(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^22(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 into :16K^2=(q+r+s-p)(p+r+s-q)(p+q+s-r)(p+q+r-s). Introducing 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 ...
, :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 Bretschneider's formula. It is a property of
cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...
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 in ...
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
cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...
, according to
Ptolemy's theorem In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematici ...
, and the formula of Coolidge reduces to Brahmagupta's formula.


Related theorems

*
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 , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century ...
for the area of a
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
is the special case obtained by taking . * The relationship between the general and extended form of Brahmagupta's formula is similar to how the
law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
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 opposit ...
. * 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