Question

Four equal circles are described about the four corners of a square so that each touches two of the others, as shown in the figure. Find the area of the shaded region, if each side of the square measures 14 cm.

Answer 1

Side of the square = 14 cm
Radius of the circle $=\frac{14}{2}$= 7 cm
Area of the quadrant of one circle = $\frac{1}{4}{\mathrm{\pi r}}^2$
$$\begin{gathered} =\frac{1}{4}\times \frac{22}{7}\times 7\times 7 \\ =38.5{\mathrm{cm}}^2 \end{gathered}$$
Area of the quadrants of four circles = $38.5\times 4=154{\mathrm{cm}}^2$

Now,
Area of the square = ${\left(\mathrm{Side}\right)}^2$
$$\begin{gathered} ={14}^2 \\ =196{\mathrm{cm}}^2 \end{gathered}$$

Area of the shaded region = Area of the square $-$ Area of the quadrants of four circles
= 196 $-$ 154
= 42 cm²