# Puzzles

## Equal areas

An equilateral triangle and a square have the same area. What is the ratio of the perimeter of the triangle to the perimeter of the square?

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Let \(A\) be the area of the square (and the triangle).

The length of a side of the square is \(\sqrt{A}\), so the perimeter of the square is \(4\sqrt{A}\).

Let \(l\) be the length of a side the triangle. Then \(\frac{1}{2}l^2\sin{60}=A\), so \(l^2=\frac{4A}{\sqrt{3}}\). Therefore \(l=\frac{2\sqrt{A}}{3^\frac{1}{4}}\) and the perimeter of the triangle is \(\frac{6\sqrt{A}}{3^\frac{1}{4}}\).

Hence the ratio of the perimeters is \(\frac{6\sqrt{A}}{3^\frac{1}{4}} : 4\sqrt{A}\) which simplifies to **3**^{3/4}:2

#### Extension

If an \(n\) sided regular polygon has the area \(A\), what is the length of one of its sides?