Difference between revisions of "Brahmagupta's Formula"
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| − | '''Brahmagupta's | + | '''Brahmagupta's Formula''' is a [[formula]] for determining the [[area]] of a [[cyclic quadrilateral]] given only the four side [[length]]s. |
== Definition == | == Definition == | ||
| − | Given a cyclic quadrilateral | + | Given a cyclic quadrilateral with side lengths <math>{a}</math>, <math>{b}</math>, <math>{c}</math>, <math>{d}</math>, the area <math>{K}</math> can be found as: |
| + | <cmath>K = \sqrt{(s-a)(s-b)(s-c)(s-d)}</cmath> | ||
| − | <math> | + | where <math>s=\frac{a+b+c+d}{2}</math> is the [[semiperimeter]] of the quadrilateral. |
| − | + | ===Proof=== | |
| + | {{incomplete|proof}} | ||
| + | |||
== Similar formulas == | == Similar formulas == | ||
Revision as of 13:33, 6 August 2008
Brahmagupta's Formula is a formula for determining the area of a cyclic quadrilateral given only the four side lengths.
Definition
Given a cyclic quadrilateral with side lengths 
, 
, 
, 
, the area 
can be found as:
![\[K = \sqrt{(s-a)(s-b)(s-c)(s-d)}\]](http://latex.artofproblemsolving.com/1/a/3/1a3a0e62995e8ed2d8e6e8a119206ee174dfdb18.png)
where 
is the semiperimeter of the quadrilateral.
Proof
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 formula reduces it to Brahmagupta's formula.
Brahmagupta's formula reduces to Heron's formula by setting the side length 
.
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