Hero's Formula - or how to find a triangle's area, given the lengths of its sides

Suppose that the sides of a triangle ABC - that is the lengths a, b and c - are given. We can use the cosine rule to derive

For the area we have the formula

Along with this we know that for a right-angled triangle with sides x, y and z and right-angle at Z

This relationship holds more generally, for in the case of an obtuse angle


so we have
whether the angle be acute (or right-angled) or obtuse, since on squaring the sign distinction disappears. Hence for our angle A we have

whether theta be acute or obtuse, so

The numerator in the square root is a difference of squares, so we have
Let now
(where s is the so-called semi-perimeter) so that the formula finally reaches the form
We have found Hero's Formula which will evaluate the area of a triangle given the lengths of its sides!
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