1. **Problem statement:** Given a large circle with diameter 12 cm and a shaded annulus area of 50 cm², find the diameter of the smaller inner circle. 2. **Formula and explanation:** The area of an annulus (ring) formed by two concentric circles is the difference of their areas: $$\text{Area}_{\text{annulus}} = \pi R^2 - \pi r^2 = \pi (R^2 - r^2)$$ where $R$ is the radius of the larger circle and $r$ is the radius of the smaller circle. 3. **Known values:** - Diameter of large circle = 12 cm, so radius $R = \frac{12}{2} = 6$ cm. - Area of annulus = 50 cm². 4. **Set up the equation:** $$50 = \pi (6^2 - r^2) = \pi (36 - r^2)$$ 5. **Solve for $r^2$:** $$\frac{50}{\pi} = 36 - r^2$$ $$r^2 = 36 - \frac{50}{\pi}$$ 6. **Calculate $r$:** $$r = \sqrt{36 - \frac{50}{\pi}}$$ Using $\pi \approx 3.1416$: $$r = \sqrt{36 - \frac{50}{3.1416}} = \sqrt{36 - 15.9155} = \sqrt{20.0845} \approx 4.48 \text{ cm}$$ 7. **Find the diameter of the smaller circle:** $$\text{Diameter} = 2r = 2 \times 4.48 = 8.96 \text{ cm}$$ **Final answer:** The diameter of the smaller circle is approximately 8.96 cm.