If a right circular cone of height 24 cm has a volume of 1232 cm³, then the area of its curved surface is

### Concept & Cone Volume and Surface Area Use the given volume to find the base radius, then use the radius and height to find the slant height. Finally, calculate the curved surface area. $$ \text{Volume} = \frac{1}{3}\pi r^2 h $$ $$ \text{CSA} = \pi r l $$ ### Step-by-Step Solution 1. **Given:** Height ($h$) = 24 cm, Volume ($V$) = 1232 cm³. 2. **Find Radius ($r$):** $\frac{1}{3} \times \frac{22}{7} \times r^2 \times 24 = 1232$ $\frac{22}{7} \times r^2 \times 8 = 1232$ $176 \times r^2 = 8624$ $r^2 = 49 \implies r = 7 \text{ cm}$ 3. **Find Slant Height ($l$):** $l = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \text{ cm}$ 4. **Calculate CSA:** $\text{CSA} = \pi r l = \frac{22}{7} \times 7 \times 25$ $\text{CSA} = 22 \times 25 = 550 \text{ cm}^2$ ### Exam Strategy & Shortcut When you see height 24 and radius 7, instantly recall the 7-24-25 right triangle triplet. This skips the step of calculating $\sqrt{625}$. ### Common Pitfall Forgetting to divide by 3 in the volume formula of the cone, which would yield an incorrect smaller radius. ### Final Answer Therefore, the correct answer is **550 cm²**.