Question

In figure, a circle is inscribed in a square ABCD and the square is circumscribed by a circle. If the radius of the smaller circle is r cm, then the area of the shaded region in cm² is ______.

Answer 1

In figure, a circle is inscribed in a square ABCD and the square is circumscribed by a circle. If the radius of the smaller circle is r cm, then the area of the shaded region in cm² is `underlinebb(1/2 (π - 2)r^2)`.

Explanation:

Given: A circle of radius r is inscribed in square ABCD; the square is circumscribed by a larger circle. So shaded region = area inside larger circle but outside the square.

Step-wise calculation:

1. Side of square, s = diameter of smaller circle = 2r.

2. Radius of larger (circumscribing) circle,

R = half the diagonal

= `(ssqrt(2))/2` 

= `(2r·sqrt(2))/2`

= `rsqrt(2)`

3. Area of larger circle = πR² 

= `π(rsqrt(2))^2`

= 2πr²

4. Area of square = s²

= (2r)²

= 4r²

5. Shaded area = (Area of larger circle) – (Area of square)

= 2πr² – 4r²

= 2r²(π – 2)

The shaded area = 2r²(π – 2) cm².