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)

Answer: Option A

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
$$\begin{array}{rl}& =4\times \frac{1}{4}\times \pi r^2\\ & =\pi r^{2}c{m}^2\end{array}$$
Area of the square :
$$\begin{array}{rl}& \\ & ={\left(2r\right)}^2\\ & =4r^{2}c{m}^2\end{array}$$
Now, according to the question,
Remaining area of the square
$$\begin{array}{rl}& 4r^2-\pi r^2=168\\ & \Rightarrow r^2\left(4-\frac{22}{7}\right)=168\\ & \Rightarrow r^2\times \left(28-22\right)=168\times 7\\ & \Rightarrow r^2=\frac{168\times 7}{6}\\ & \Rightarrow r^2=28\times 7=7\times 4\times 7\\ & \Rightarrow r=\sqrt{7\times 7\times 2\times 2}\\ & \Rightarrow r=7\times 2\\ & \Rightarrow r=14cm\end{array}$$