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In geometry, Heron's formula (sometimes given as Hero's
formula) states that the area S of a triangle whose sides have lengths a, b, c is given by
-
where
-
(see also square root).
History
The formula is credited to Heron of Alexandria in the
1st century A.D., and a proof can be found in his book Metrica. It
is now believed that Archimedes already knew the formula, and it is of course
possible that it has been known long before.
Proof
A modern proof, which uses algebra and trigonometry and is quite unlike the one provided by Heron, follows. Let a, b, c be
the sides of the triangle and A, B, C the angles opposite
those sides. We have
-
by the law of cosines. From this we get with some algebra
- .
The altitude of the triangle on base a has
length bsin(C), and it follows
-
-
-
-
Here the somewhat tedious but simple algebra in the last step was omitted.
Generalizations
The formula is in fact a special case of Brahmagupta's
formula for the area of a cyclic quadrilateral; both of
which are special cases of Bretschneider's formula for the area of a quadrilateral.
Expressing Heron's formula with a determinant in terms of the squares of
the distances between the three given vertices,
-
illustrates its similarity to Tartaglia's formula for
the volume of a four-simplex.
See also
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