Compute the length of an equilateral triangle's side given the area?

Question

Given the area of an equilateral triangle, what is an algorithm to determine the length of a side?

Answer 1

Let $s$ be the side, and $A$ the area. Drop a perpendicular from one vertex to the opposite side. By the Pythagorean Theorem, the height of the triangle is $\sqrt{s^2-\frac{1}{4}s^2}=\frac{s\sqrt{3}}{2}$. It follows that $$A=\frac{s^2\sqrt{3}}{4}.$$ Thus $$s^2=\frac{4A}{\sqrt{3}},$$ and therefore $$s=\sqrt{\frac{4A}{\sqrt{3}}}.$$ There are several ways to rewrite the above expression.