A hemisphere and a cone have equal bases. If their heights are also equal, then the ratio of their curved surfaces will be

### Concept & Curved Surface Area When a hemisphere and a cone have equal bases and heights, the height of the cone equals the radius of their common base. We compare their standard curved surface area (CSA) formulas. CSA of hemisphere: $$2\pi r^2$$ CSA of cone: $$\pi r l$$ Where $l$ (slant height) = $$\sqrt{r^2 + h^2}$$ ### Step-by-Step Solution 1. Let the common base radius be $r$. 2. The height of the hemisphere is inherently its radius $r$. Thus, the height of the cone is also $h = r$. 3. Calculate the slant height $l$ of the cone: $l = \sqrt{r^2 + h^2} = \sqrt{r^2 + r^2} = \sqrt{2r^2} = \sqrt{2}r$. 4. Find the CSA of the hemisphere: $CSA_{hemi} = 2\pi r^2$. 5. Find the CSA of the cone: $CSA_{cone} = \pi r l = \pi r (\sqrt{2}r) = \sqrt{2}\pi r^2$. 6. Determine the ratio: $\frac{CSA_{hemi}}{CSA_{cone}} = \frac{2\pi r^2}{\sqrt{2}\pi r^2} = \frac{2}{\sqrt{2}}$. 7. Rationalize the denominator by multiplying top and bottom by $\sqrt{2}$: $\frac{2\sqrt{2}}{2} = \sqrt{2} : 1$. ### Exam Strategy & Shortcut Memorize that for a cone with $h=r$, the slant height is always $r\sqrt{2}$. The CSA is simply $\pi r(r\sqrt{2}) = \sqrt{2}\pi r^2$. Comparing this to the hemisphere's $2\pi r^2$, the $\pi r^2$ terms cancel immediately, leaving $2 : \sqrt{2}$, which simplifies to $\sqrt{2} : 1$. ### Common Pitfall A frequent error is assuming the CSA of a cone with $h=r$ is comparable linearly to its height, ignoring the slant height calculation ($l$). Always use $l$ for cone surface areas. ### Final Answer Therefore, the correct answer is **$\sqrt{2}$ : 1**.