3320.11 – A Ratio of Areas

Find the ratio of the area of the circle circumscribed about a given square to the area of the circle inscribed in that same square.

Solution

We want the ratio of the area $A$ of the large circle to that $a$ of the small circle in the diagram below. We see that $A = \pi R^2$ $a = \pi r^2$ , so that

$\frac{\pi R^2}{\pi r^2} = \frac{R^2}{r^2} = \frac{(r \sqrt{2})^2}{r^2} = \sqrt{2}^2 = 2.$

Surprising?