Trigonometry Proof of

Heron's Formula



In any triangle, the altitude to a side is equal to the product of the sine of the angle subtending the altitude and a side from the angle to the vertex of the triangle.


In this picutre, the altitude to side c is    b sin A    or  a sin B

(Setting these equal and rewriting as ratios leads to the demonstration of the Law of Sines)

Since in the triangle pictured     we have 


Now we look for a substitution for sin A in terms of a, b, and c. It is readily (if messy) available from the Law of Cosines

and substitution in the identity

Factor (easier than multiplying it out) to get

Rewrite with common denominators.


Factor and rearrange

Now where the semiperimeter s is defined by

the four expressions under the radical are 2s, 2(s - a), 2(s - b), and 2(s - c).   So


we have