A square is inscribed in a circle whose radius is 4 cm. The area of the portion between the circle and the square is :

### Concept & Area of Inscribed Square To find the area between a circle and an inscribed square, we subtract the square's area from the circle's area. For a square inscribed in a circle, its diagonal equals the circle's diameter. $$ \text{Area of circle} = \pi r^2 $$ $$ \text{Area of square} = \frac{1}{2} \times d^2 $$ ### Step-by-Step Solution * Given: The radius of the circle $r = 4$ cm. * The area of the circle is $\pi(4)^2 = 16\pi$ cm$^2$. * The diagonal of the inscribed square is equal to the diameter of the circle. * Diameter $d = 2 \times 4 = 8$ cm. * The area of the square is $\frac{1}{2} \times (8)^2 = \frac{1}{2} \times 64 = 32$ cm$^2$. * The area of the portion between the circle and the square is $16\pi - 32$. ### Exam Strategy & Shortcut Remember that for a square inscribed in a circle of radius $r$, its area is always $2r^2$. Thus, the shaded area formula is simply $r^2(\pi - 2)$. Plugging in $r = 4$ yields $16(\pi - 2) = 16\pi - 32$ instantly. ### Common Pitfall A common mistake is treating the radius of the circle as the side of the square or half the side, rather than half the diagonal. Always draw a quick sketch to confirm geometry relationships. ### Final Answer Therefore, the correct answer is **$(16\pi - 32)$**.