Question

Four identical circles are drawn taking the vertices of a square as centers. The circles are tangential to one another. Another circle is drawn so that it is tangential to all the circles and lies within the square. Find the ratio of the sum of the areas of the four circles lying within the square to that of the smaller circle?

• A. 2/(1-√2)
• B. 2/(1-√2)²
• C. 1/(√2-1)²
• D. 1/(√2+1)²

Answer 1

The correct option is C 1/(√2-1)²

Let the radius of the identical circles be r. Hence, Areas of the four circles lying within the square = 4 x $\pi$ x $\frac{r^2}{4}$ = $\pi$ x $r^2$

Radius of the smaller circle = $\frac{(diagonalofsquare-2r)}{2}$ = $\frac{(2\sqrt{2}r-2r)}{2}$ = r ($\sqrt{2}$-1)

Required ratio = $\frac{\pi x{r}^2}{\pi x\left[x^2\right(\sqrt{2}-1{)}^2]}$ = $\frac{1}{(\sqrt{2}-1{)}^2}$

Shortcut : Lateral Thinking

The question is basically the ratio areas of larger to smaller circle because sum of areas within the square of the 4 circles add up to a full large circle.

Approximately compare the radii of small and large circles; the ratio is nearly 3.5/1, square of which is approx. 6/1. Only option c) works. This is a good approach which can be used for a lot of geometry questions.