Heron's formula
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\n:
where
-
(see also
square root).
History
The formula is credited to Heron of Alexandria in the 1st century AD, 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\n:\nby the law of cosines. From this we get with some algebra\n:.\nThe altitude of the triangle on base a has length bsin(C), and it follows\n:\n:\n:\n:\nHere 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,\n:\nillustrates its similarity to Tartaglia's formula for the volume of a four-simplex.
See also
\n*Synthetic geometry
Category:Euclidean geometry\nCategory:Theorems\nCategory:Triangles
\n
