Question

Four equal circles are described about the four corners of a square so that each circle touches two of the others. Find the area of the space enclosed between the circumferences of the circles, each side of the square measuring $24cm$.

Answer 1

The side of square is given by $24cm$

Area of Square is $a^2={24}^2=576{cm}^2$

The radius of the $4$ circles at corners of square is given by

$\frac{24}{2}=12cm$

The area of quadrant of circle inside the square is $\frac{1}{4}\times \pi \times r^2$

Since there are $4$ such quadrants so area = $4\times \frac{1}{4}\pi r^2=\pi r^2$

Area of 4 quadrants$=\pi \left(12\right)\left(12\right)=144\pi =3.14\left(144\right)=452.16{cm}^2$

So the area of the space enclosed between the circumferences of the circles = Area of Square - Area of 4 quadrants

= $(576-452.16){cm}^2=123.84{cm}^2$