Difference between revisions of "Brahmagupta's Formula"

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'''Brahmagupta's formula''' is a [[formula]] for determining the [[area]] of a [[cyclic quadrilateral]] given only the four side lengths.
 
'''Brahmagupta's formula''' is a [[formula]] for determining the [[area]] of a [[cyclic quadrilateral]] given only the four side lengths.
  
=== Definition ===
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== Definition ==
  
 
Given a cyclic quadrilateral has side lengths <math>{a}, {b}, {c}, {d}</math>, the area <math>{K}</math> can be found as:
 
Given a cyclic quadrilateral has side lengths <math>{a}, {b}, {c}, {d}</math>, the area <math>{K}</math> can be found as:
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where the [[semiperimeter]] <math>s=\frac{a+b+c+d}{2}</math>.
 
where the [[semiperimeter]] <math>s=\frac{a+b+c+d}{2}</math>.
  
=== Similar formulas ===
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== 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.
 
[[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 <math>{d}=0</math>.
 
Brahmagupta's formula reduces to [[Heron's formula]] by setting the side length <math>{d}=0</math>.
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{{stub}}
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[[Category:Geometry]]
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[[Category:Theorem]]

Revision as of 17:01, 7 October 2007

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 has side lengths ${a}, {b}, {c}, {d}$, the area ${K}$ can be found as:

$K = \sqrt{(s-a)(s-b)(s-c)(s-d)}$

where the semiperimeter $s=\frac{a+b+c+d}{2}$.

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 ${d}=0$.

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