A square and an equilateral triangle have the property that the area of each is the perimeter of the other. What is the area of the square?
Assuming the square has side length $x$, it has area $x^{2}$, so the equilateral triangle has side length $x^{2} / 3$. The area of the equilateral triangle is then given in two ways by

$$
4 x=\sqrt{3} / 4 \cdot\left(\frac{x^{2}}{3}\right)^{2} .
$$

Solving gives $x^{3}=\frac{144}{\sqrt{3}}=48 \sqrt{3}$. Then $x^{6}=48^{2} \cdot 3$ and $x^{2}=\sqrt[3]{48^{2} \cdot 3}=\sqrt[3]{2^{8} 3^{3}}=$ $12 \sqrt[3]{4}$