In yesterday’s post, we discussed how the area of a triangle can be found using SAS: two sides and the angle between the two sides. We found that
This can be used as the starting point for the derivation of Heron’s formula, which determines the area of a triangle using SSS (i.e., only the three sides). I won’t give the full derivation in this post — there’s no point in me retyping the details — but will refer to the Wikipedia page and the MathWorld page for the details. However, I will give the big ideas behind the derivation.
1. We begin by recalling that . Since , we know that must be positive, so that
2. From the Law of Cosines, we know that
3. Substituting, we see that
4. This last expression only contains the side lengths , , and . So the “only” work that’s left is simplifying this right-hand side and seeing what happens. After considerable work — requiring factoring the difference of two squares on two different steps — we end up with Heron’s formula:
where is the semiperimeter, or half the perimeter of the triangle.
A final note: If you actually are able to start with Step 3 and end with Heron’s formula on your own — without consulting a textbook or the Internet if you get stuck — feel free to cry out “More power!” and grunt like Tim “The Toolman” Taylor: