Question

The ratio of the total surface area of a sphere and that of a hemisphere having the same radius is

• A. 2 : 1
• B. 4 : 9
• C. 3 : 2
• D. 4 : 3

Hint:

This is a direct comparison of formulas. Make sure you use the total surface area of the hemisphere (\(3\pi R^2\)), not just its curved surface area (\(2\pi R^2\)).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We need to find the ratio of the total surface area (TSA) of a sphere to the TSA of a hemisphere, given that they both have the same radius, \(R\).

Step 2: Key Formula or Approach:
1. The total surface area of a sphere of radius \(R\) is \(TSA_{sphere} = 4\pi R^2\).
2. The total surface area of a hemisphere of radius \(R\) is the sum of its curved area (\(2\pi R^2\)) and its circular base area (\(\pi R^2\)), which is \(TSA_{hemisphere} = 3\pi R^2\).
3. We need to find the ratio \(\frac{TSA_{sphere}}{TSA_{hemisphere}}\).

Step 3: Detailed Explanation:
\[ \text{Ratio} = \frac{TSA_{sphere}}{TSA_{hemisphere}} = \frac{4\pi R^2}{3\pi R^2} \] Cancel the common terms \(\pi R^2\) from the numerator and the denominator:
\[ \text{Ratio} = \frac{4}{3} \] So, the ratio is 4 : 3.

Step 4: Final Answer:
The ratio is 4 : 3.