Discussion
An equilateral triangle, a square and a circle have equal areas.
What is the ratio of the perimeters of the equilateral triangle to square to circle?
Answer: B
The ratio is \(\sqrt{3\sqrt{3}} : 2 : \sqrt{\pi}\).
Give each figure area 1. For the equilateral triangle of side \(s\), \(\frac{\sqrt{3}}{4}s^{2} = 1\), so \(s^{2} = 4/\sqrt{3}\) and its perimeter is \(3s = \sqrt{36/\sqrt{3}} = 2\sqrt{3\sqrt{3}}\). The square has side 1 and perimeter 4. The circle has \(\pi r^{2} = 1\), so its perimeter is \(2\pi r = 2\sqrt{\pi}\).
\[ \begin{aligned} &2\sqrt{3\sqrt{3}} : 4 : 2\sqrt{\pi} \\ &= \sqrt{3\sqrt{3}} : 2 : \sqrt{\pi} \end{aligned} \]
The option \(\sqrt{3\sqrt{3}} : 4 : 2\sqrt{\pi}\) halves only the triangle's term, and the one ending in \(2\sqrt{\pi}\) halves only the square's. \(3\sqrt{3} : 2 : \sqrt{\pi}\) forgets the outer square root.
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