The area of an equilateral triangle is numerically equal to the length of one of its sides. What is the perimeter of the triangle, in units? Express your answer in simplest radical form.
Let's call the length of one of the sides of the equilateral triangle x. In an equilateral triangle, all sides are equal, so the perimeter would be 3x. Let's call the height of the triangle h. We know that the area of the triangle is $xh/2$. And we know that the area is also x. So $xh/2=x$. That means that $h/2=1$, so $h=2$. Now we just need to find the height in terms of x. We can use the Pythagorean theorem, since the triangle can be split into two 30-60-90 triangles. So $x^2=h^2+(x/2)^2$. Simplifying that gives us $x^2=4+x^2/4$. Which is the same as $3x^2/4=4$. So $x^2=16/3$. And $x=4/\sqrt{3}$. So the perimeter is $3x=4\sqrt{3}$. Right.

