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 area 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. areas of the four circles lying within the square = 4 x $\pi$ x $r^2$ = 4 x $\pi$ x $r^2$

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

Area of smaller circle = $\pi$ x $r^2$ x ${(\sqrt{2}-1)}^2$

Required ratio = $\frac{4}{(\sqrt{2}-1{)}^2}$