1. **State the problem:** We have a cone with base diameter 20 cm and slant height 25 cm. A circle is drawn on the cone's surface at 10 cm above the base along the slant height. We need to find the curved surface area of the cone that is *not* painted grey, i.e., the curved surface area below the circle. 2. **Relevant formula:** The curved surface area (CSA) of a cone is given by $$\text{CSA} = \pi r l$$ where $r$ is the radius of the base and $l$ is the slant height. 3. **Given:** - Base diameter = 20 cm, so base radius $r = \frac{20}{2} = 10$ cm - Total slant height $L = 25$ cm - Circle drawn at slant height $l_1 = 10$ cm from the base 4. **Find radius of the circle at slant height 10 cm:** The radius of the cone's cross-section varies linearly from 0 at the apex to 10 cm at the base (slant height 25 cm). So radius at slant height $l_1$ is: $$r_1 = \frac{l_1}{L} \times r = \frac{10}{25} \times 10 = 4 \text{ cm}$$ 5. **Calculate curved surface area below the circle:** The curved surface area from the base up to slant height $l_1$ is: $$\text{CSA}_{\text{below}} = \pi r_1 l_1 = \pi \times 4 \times 10 = 40\pi$$ 6. **Final answer:** The curved surface area of the cone that is not painted grey is $$\boxed{40\pi} \text{ cm}^2.$$