In geometry, Brahmagupta's formula finds the area of any quadrilateral given the lengths of the sides and some of their angles. In its most common form, it yields the area of quadrilaterals that can be inscribed in a circle.
Basic form Edit
In its basic and easiest-to-remember form, Brahmagupta's formula gives the area of a cyclic quadrilateral whose sides have lengths a, b, c, d as
where s, the semiperimeter, is
The assertion that the area of the quadrilateral is given by Brahmagupta's formula is equivalent to the assertion that it is equal to
Brahmagupta's formula may be seen as a formula in the half-lengths of the sides, but it also gives the area as a formula in the altitudes from the center to the sides, although if the quadrilateral does not contain the center, the altitude to the longest side must be taken as negative.
Proof of Brahmagupta's formula Edit
Area of the cyclic quadrilateral = Area of + Area of
But since is a cyclic quadrilateral, Hence Therefore
Applying law of cosines for and and equating the expressions for side we have
Substituting (since angles and are supplementary) and rearranging, we have
Substituting this in the equation for area,
which is of the form and hence can be written in the form as
Taking square root, we get
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 pair 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.) It follows from this fact that the area of a cyclic quadrilateral is the maximum possible area for any quadrilateral with the given side lengths.
where p and q are the lengths of the diagonals of the quadrilateral.
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
giving the basic form of Brahmagupta's formula.
Related theorems Edit
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