2007 SMT Geometry Round Problem 1

Problem

An equilateral triangle has perimeter numerically equal to its area, which is not zero. Find its side length.

Solution

Let's say the side length of this equilateral triangle $s$ . Then, its perimeter must be $3s$ . By the area formula for an equilateral triangle known its side length, its area must be $\frac{s^2\sqrt3}{4}$ . Because these two are numerically equal, we have $3s=\frac{s^2\sqrt3}{4}$ , so $12s=s^2\sqrt3$ . Because $s$ has to be nonzero (if it is zero, then the area and perimeter would be zero), we can divide by $s$ on both sides to get $12=s\sqrt3$ , so $\frac{12}{\sqrt3}=s$ , so $s=\frac{12\sqrt3}{\sqrt3\times\sqrt3}$ , so $s=4\sqrt3$ . Therefore, our answer is $\boxed{\mathrm{4\sqrt3}}$ .

~Yuhao2012

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2007 SMT Geometry Round Problem 1

Problem

Solution