If the ratio between the perimeters of two squares is , find the ratio between their areas.
Denote the side length of the smaller square by and the side length of the larger square by . Then the perimeters are given by and .
It therefore follows that
which can be rearranged to
The area of the smaller square is and the area of the larger square is . The ratio of the areas is therefore
Substituting the first order equation into the expression for the ratio of the areas gives
Therefore, the ratio of the areas is .