Heron's Formula

\mathrm{area} = \sqrt{s\left(s-a
ight)\left(s-b
ight)\left(s-c
ight)}\,

where ''s'' is the triangle's semiperimeter:

:

s=rac{a+b+c}{2}

(see also Square Root ). Heron's formula can also be written

:

\mathrm{area}={\ \sqrt{(a+b+c)(a+b-c)(b+c-a)(c+a-b)\,}\ \over 4}.\,

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 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 Angle s opposite those sides. We have
:

\cos(C) = rac{a^2+b^2-c^2}{2ab}

by the Law Of Cosines . From this we get with some algebra
:

\sin(C) = \sqrt{1-\cos^2(C)} = rac{\sqrt{4a^2 b^2 -(a^2 +b^2 -c^2)^2 }}{2ab}

.
The Altitude of the triangle on base ''a'' has length ''b''sin(C), and it follows

Here the simple 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'' ≥ ''c''
and computing
:

S = rac{1}{4}\sqrt{(a+(b+c)) (c-(a-b)) (c+(a-b)) (a+(b-c))}

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,
:

S =  rac{1}{4} \sqrt{ \begin{vmatrix} 
  0 & a^2 & b^2 & 1 \
a^2 & 0   & c^2 & 1 \
b^2 & c^2 & 0   & 1 \
  1 &   1 &   1 & 0
\end{vmatrix} }

illustrates its similarity to Tartaglia's Formula for the Volume of a Four-simplex .

SEE ALSO

Synthetic Geometry

Heronian Triangle

EXTERNAL LINKS

MathWorld entry on Heron's Formula

Semiperimeter, incircle and excircles of a triangle by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas".

A Proof of the Pythagorean Theorem From Heron's Formula at Cut-the-knot

REFERENCES

Information About

Heron's Formula