1. **Stating the problem:** We are given three geometric figures with various side lengths labeled. The first figure is a cone with a slanted side of 30 cm and a base diameter of 20 cm. 2. **Formula used:** To find the height of a cone when the slant height ($l$) and base radius ($r$) are known, we use the Pythagorean theorem: $$h = \sqrt{l^2 - r^2}$$ where $h$ is the height, $l$ is the slant height, and $r$ is the radius of the base. 3. **Applying the formula:** - The base diameter is 20 cm, so the radius $r = \frac{20}{2} = 10$ cm. - The slant height $l = 30$ cm. 4. **Calculate the height:** $$h = \sqrt{30^2 - 10^2} = \sqrt{900 - 100} = \sqrt{800}$$ 5. **Simplify the square root:** $$\sqrt{800} = \sqrt{16 \times 50} = 4\sqrt{50} = 4 \times 5 \sqrt{2} = 20\sqrt{2}$$ 6. **Final answer:** The height of the cone is $$h = 20\sqrt{2} \text{ cm} \approx 28.28 \text{ cm}$$