A right angled sector of radius r cm is rolled up into a cone in such a way that the two binding radii are joined together. Then the curved surface area of the cone is

Options

A. `pi r^(2) cm^(2)`
B. `4 pir^(2) cm^(2)`
C. `(pi r^(2))/(4) cm^(2)`
D. `2pi r^(2) cm^(2)`

Answer

To solve the problem of finding the curved surface area of a cone formed by rolling a right-angled sector of radius \( r \) cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Right-Angled Sector**:
- A right-angled sector is a quarter of a circle. The radius of the sector is given as \( r \) cm.

2. **Determine the Area of the Right-Angled Sector**:
- The area of a full circle is given by the formula \( \pi r^2 \).
- Since we have a quarter circle (right-angled sector), we need to take one-fourth of the area of the full circle:
\[ \text{Area of the right-angled sector} = \frac{1}{4} \times \pi r^2 = \frac{\pi r^2}{4} \]

3. **Rolling the Sector into a Cone**:
- When the right-angled sector is rolled into a cone, the two radii of the sector become the slant height (l) of the cone, and the arc length of the sector becomes the circumference of the base of the cone.

4. **Calculate the Arc Length**:
- The angle of the right-angled sector is \( 90^\circ \) or \( \frac{\pi}{2} \) radians.
- The arc length (which becomes the circumference of the base of the cone) can be calculated as:
\[ \text{Arc Length} = \frac{\theta}{2\pi} \times 2\pi r = \frac{90}{360} \times 2\pi r = \frac{1}{4} \times 2\pi r = \frac{\pi r}{2} \]

5. **Relate the Arc Length to the Base Radius of the Cone**:
- Let the radius of the base of the cone be \( R \). The circumference of the base of the cone is given by:
\[ 2\pi R = \frac{\pi r}{2} \]
- Solving for \( R \):
\[ R = \frac{r}{4} \]

6. **Curved Surface Area of the Cone**:
- The formula for the curved surface area (CSA) of a cone is given by:
\[ \text{CSA} = \pi R l \]
- Here, \( l = r \) (the slant height) and \( R = \frac{r}{4} \):
\[ \text{CSA} = \pi \left(\frac{r}{4}\right) r = \frac{\pi r^2}{4} \]

### Final Answer:
The curved surface area of the cone formed by rolling the right-angled sector is:
\[ \frac{\pi r^2}{4} \text{ cm}^2 \]