1. **Problem Statement:** Calculate the area of: i. the circle with radius $r = 3.5$ cm ii. the square QPQR inscribed in the circle iii. the shaded region (area of circle minus area of square) 2. **Formulas and Rules:** - Area of a circle: $$A_{circle} = \pi r^2$$ - Area of a square: $$A_{square} = s^2$$ where $s$ is the side length - The square QPQR is inscribed in the circle, so its diagonal equals the diameter of the circle. - Diameter of circle: $$d = 2r$$ - Diagonal of square: $$d = s\sqrt{2}$$ 3. **Calculate the area of the circle:** $$r = 3.5$$ $$\pi = \frac{22}{7}$$ $$A_{circle} = \pi r^2 = \frac{22}{7} \times (3.5)^2 = \frac{22}{7} \times 12.25$$ $$= 22 \times \frac{12.25}{7} = 22 \times 1.75 = 38.5$$ 4. **Calculate the side length of the square:** Diameter of circle: $$d = 2r = 2 \times 3.5 = 7$$ Diagonal of square equals diameter: $$d = s\sqrt{2}$$ Solve for $s$: $$s = \frac{d}{\sqrt{2}} = \frac{7}{\sqrt{2}}$$ Rationalize denominator: $$s = \frac{7}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{7\sqrt{2}}{2}$$ 5. **Calculate the area of the square:** $$A_{square} = s^2 = \left(\frac{7\sqrt{2}}{2}\right)^2 = \frac{49 \times 2}{4} = \frac{98}{4} = 24.5$$ 6. **Calculate the shaded region area:** $$A_{shaded} = A_{circle} - A_{square} = 38.5 - 24.5 = 14$$ **Final answers:** - Area of the circle = 38.5 cm$^2$ - Area of the square QPQR = 24.5 cm$^2$ - Area of the shaded region = 14 cm$^2$