Question

At the beginning of a carrom game the coins are arranged in the central circle of board as shown. What is the area in (square inches in nearest integer) of the central circle not occupied by the coins if the radius of each coins is $\frac{1}{2}$ an inch

Answer 1

Radius of each coin $=\frac{1}{2}$ inch

and from figure, we can say that

Radius of central circle $=5\times$ radius of each coin

Radius of central circle $5\times \frac{!}{2}$

$=\frac{5}{2}$ inch

And as the picture shows

Total number of coins in the central circle $=19$

Hence

Area of circle occupied by the coins $=19\times$area occupied by oen coin$=19\times \pi R^2$

$=19\times \pi \times {\left(\frac{1}{2}\right)}^2$

$=\frac{19}{4}\pi$ sq. inches

Now, total area of central circle $=\pi \times {\left(\frac{5}{2}\right)}^2$

$\pi \times \frac{25}{4}$

ARea of central circle not occupied by coins $=\frac{25}{4}\pi -\frac{19}{4}\pi$

$=\frac{6\pi }{4}$

$=4.71$ sq. inces