Heron's formula
In
geometry, Heron's formula states that the
area of a
triangle whose sides have lengths
a,
b and
c is
where
s is the triangle's semiperimeter:
. Heron's formula can also be written
Encyclopedia
In
geometry,
Heron's formula states that the
area of a
triangle whose sides have lengths
a,
b and
c is
where
s is the triangle's
semiperimeter:
. Heron's formula can also be written
History
The formula is credited to
Heron of Alexandria in the
1st century, 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 was 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, and it follows
Here the algebra in the last step was omitted.
Numerical stability
Heron's formula as given above is
numerically unstable for triangles with a very small angle.
A stable alternative involves arranging the lengths of the sides so that:
a ≥
b ≥
cand computing
The brackets in the above formula are required in order to prevent numerical instability in the evaluation.
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
References
External links
- by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas".
- at cut-the-knot
