The ratio of the surface area of a sphere and the curved surface area of the cylinder circumscribing the sphere is

### Concept & Circumscribed Geometry When a cylinder circumscribes a sphere, it perfectly encloses it. This means the cylinder's radius is equal to the sphere's radius, and the cylinder's height is equal to the sphere's diameter. $$Surface Area_{sphere} = 4\pi r^2$$ $$Curved Surface Area_{cylinder} = 2\pi rh$$ ### Step-by-Step Solution 1. Let the radius of the sphere be $r$. The surface area of the sphere is $4\pi r^2$. 2. For the circumscribing cylinder, the base radius $R = r$. 3. The height of the circumscribing cylinder $h$ is exactly the diameter of the enclosed sphere, so $h = 2r$. 4. Calculate the curved surface area (CSA) of the cylinder: $CSA = 2\pi R h = 2\pi (r) (2r) = 4\pi r^2$. 5. Find the required ratio: $\frac{Surface Area_{sphere}}{Curved Surface Area_{cylinder}} = \frac{4\pi r^2}{4\pi r^2} = \frac{1}{1}$. 6. The ratio is $1 : 1$. ### Exam Strategy & Shortcut This is a standard geometric theorem formulated by Archimedes. Memorize the fact that a sphere and its perfectly circumscribing cylinder have exactly the same surface area (ratio 1:1) and their volumes are in a 2:3 ratio. ### Common Pitfall Confusing the "curved surface area" with the "total surface area" of the cylinder. If total surface area were used ($2\pi r(r+h) = 6\pi r^2$), the ratio would be 4:6 or 2:3. ### Final Answer Therefore, the correct answer is **1 : 1**.