, Heron's formula
states that the area S
of a triangle
whose sides have lengths a
is given by
(see also square root
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.
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
of the triangle on base a
has length b
sin(C), and it follows
Here the somewhat tedious but simple algebra in the last step was omitted.
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