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

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If you know the length of a side of an equilateral triangle, do you know how to find the area? –  Jonas Meyer Nov 27 '12 at 6:43
    
Yeah, I've found the algorithms for it, but I'm hideous at working out the inverse of algorithms. If I have an answer in front of me I'll be able to visualise how it works :) Never learnt geometry so this is all a bit new to me. –  Djentleman Nov 27 '12 at 6:44
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If $A = C\cdot s^2$, then $s=\sqrt{\frac{1}{C}\cdot A}$. –  Jonas Meyer Nov 27 '12 at 6:45
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1 Answer

up vote 4 down vote accepted

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.

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