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

~Yuhao2012