Question

In the figure, when the outer circles all have radii 'r', then the radius of the inner circle will be

• A. $\sqrt{2}r$
• B. $(\sqrt{2}-1)r$
• C. $\frac{1}{\sqrt{2}r}$
• D. $\frac{2}{(\sqrt{2}+1)r}$

Answer 1

Answer: (B)

$(\sqrt{2}-1)r$

Let the same radii of all four circles be 'r'

Let the inner circle radius be 'R'

Joining any two diagonally opposite circles center by a line passing through the center of inner circle, we get a diagonal for the square, a square is formed by joining of the centers of four outer circles.

Also we know that:

Length of side $=2\times r=2r$

Length of diagonal $=r+2R+r=2(R+r)$

Therefore, $2(R+r)=2r\sqrt{2}$

$\therefore R+r=r\sqrt{2}$

Thus, $R=r(\sqrt{2}-1)$ is the required inner circle radius r in terms of the outer circles with radius R