Lateral height of a cone is 10 cm and height is 8 cm. Curved surface area of a cone is:

Options

A. `187.5 cm^(2)`
B. `189.5 cm^(2)`
C. `188.5 cm^(2)`
D. `190.5 cm^(2)`

Answer

To find the curved surface area of the cone given the lateral height (slant height) and the vertical height, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values:**
- Lateral height (slant height) \( L = 10 \, \text{cm} \)
- Height \( H = 8 \, \text{cm} \)

2. **Use the Pythagorean theorem to find the radius \( R \):**
The relationship between the lateral height, height, and radius of the cone is given by the formula:
\[ L^2 = H^2 + R^2 \]
Substituting the known values:
\[ 10^2 = 8^2 + R^2 \]
\[ 100 = 64 + R^2 \]

3. **Solve for \( R^2 \):**
Rearranging the equation gives:
\[ R^2 = 100 - 64 \]
\[ R^2 = 36 \]

4. **Find the radius \( R \):**
Taking the square root of both sides:
\[ R = \sqrt{36} = 6 \, \text{cm} \]

5. **Calculate the curved surface area (CSA) of the cone:**
The formula for the curved surface area of a cone is:
\[ \text{CSA} = \pi R L \]
Substituting the values of \( R \) and \( L \):
\[ \text{CSA} = \frac{22}{7} \times 6 \times 10 \]

6. **Perform the multiplication:**
\[ \text{CSA} = \frac{22}{7} \times 60 \]
\[ \text{CSA} = \frac{1320}{7} \approx 188.57 \, \text{cm}^2 \]

### Final Answer:
The curved surface area of the cone is approximately \( 188.57 \, \text{cm}^2 \).