Four circles having equal radii are drawn with centres at the four corners of a square. Each circle touches the other two adjacent circles. If the remaining area of the square is 168 cm², what is the size of the radius of the circle ? (in centimetres)

Solution(By Examveda Team)

Let the radius of each circle be r cm

Then the side of the square will be 2r cm
Area covered by the four circle in the square
$$\eqalign{ & = 4 \times \frac{1}{4} \times \pi {r^2} \cr & = \pi {r^2}c{m^2} \cr} $$
Area of the square :
$$\eqalign{ & = {\left( {2r} \right)^2} \cr & = 4{r^2}c{m^2} \cr} $$
Now, according to the question,
Remaining area of the square
$$\eqalign{ & 4{r^2} - \pi {r^2} = 168 \cr & \Rightarrow {r^2}\left( {4 - \frac{{22}}{7}} \right) = 168 \cr & \Rightarrow {r^2} \times \left( {28 - 22} \right) = 168 \times 7 \cr & \Rightarrow {r^2} = \frac{{168 \times 7}}{6} \cr & \Rightarrow {r^2} = 28 \times 7 = 7 \times 4 \times 7 \cr & \Rightarrow r = \sqrt {7 \times 7 \times 2 \times 2} \cr & \Rightarrow r = 7 \times 2 \cr & \Rightarrow r = 14\,cm \cr} $$