1. **State the problem:** We need to find the area of the shaded portion formed by two semicircles with diameters 42 cm and 21 cm. 2. **Formula for the area of a semicircle:** The area of a semicircle is given by $$\text{Area} = \frac{1}{2} \pi r^2$$ where $r$ is the radius of the semicircle. 3. **Calculate the radius of each semicircle:** - For the semicircle with diameter 42 cm, radius $r_1 = \frac{42}{2} = 21$ cm. - For the semicircle with diameter 21 cm, radius $r_2 = \frac{21}{2} = 10.5$ cm. 4. **Calculate the area of each semicircle:** - Area of larger semicircle: $$A_1 = \frac{1}{2} \times \frac{22}{7} \times 21^2 = \frac{1}{2} \times \frac{22}{7} \times 441$$ Simplify: $$A_1 = \frac{1}{2} \times 22 \times 63 = 11 \times 63 = 693 \text{ cm}^2$$ - Area of smaller semicircle: $$A_2 = \frac{1}{2} \times \frac{22}{7} \times 10.5^2 = \frac{1}{2} \times \frac{22}{7} \times 110.25$$ Simplify: $$A_2 = \frac{1}{2} \times 22 \times 15.75 = 11 \times 15.75 = 173.25 \text{ cm}^2$$ 5. **Find the total shaded area:** Since the two semicircles are shaded, the total shaded area is the sum of their areas: $$\text{Total area} = A_1 + A_2 = 693 + 173.25 = 866.25 \text{ cm}^2$$ **Final answer:** The area of the shaded portion is $866.25$ cm$^2$.