# Tag: law of cosines

# Area of a triangle: SSS (Part 5)

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

,

or

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: