Brahmagupta's Formula
Brahmagupta's Formula is a formula for determining the area of a cyclic quadrilateral given only the four side lengths, given as follows: where
,
,
,
are the four side lengths and
.
Proofs
If we draw , we find that
. Since
,
. Hence,
. Multiplying by 2 and squaring, we get:
Substituting
results in
By the Law of Cosines,
.
, so a little rearranging gives
Similar formulas
Bretschneider's formula gives a formula for the area of a non-cyclic quadrilateral given only the side lengths; applying Ptolemy's Theorem to Bretschneider's amnado
Brahmagupta's formula reduces to Heron's formula by setting the side length .
A similar formula which Brahmagupta derived for the area of a general quadrilateral is
where
is the semiperimeter of the quadrilateral. What happens when the quadrilateral is cyclic?
Problems
Intermediate
is a cyclic quadrilateral that has an inscribed circle. The diagonals of
intersect at
. If
and
then the area of the inscribed circle of
can be expressed as
, where
and
are relatively prime positive integers. Determine
. (Source)
- Quadrilateral
with side lengths
is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form
where
and
are positive integers such that
and
have no common prime factor. What is
(Source)