The perimeter of a quadrant of a circle of radius 'r' is :

Options

A. `(pi r)/(2)`
B. `2 pi r`
C. `(r)/(2) [pi + 4]`
D. `2 pi r + (r)/(2)`

Answer

To find the perimeter of a quadrant of a circle with radius 'r', we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Quadrant**: A quadrant is one-fourth of a circle. It is formed by drawing two radii from the center of the circle to the circumference, creating a right angle (90 degrees).

2. **Identify the Components of the Perimeter**: The perimeter of the quadrant consists of:
- The two straight sides (the radii) which are both equal to 'r'.
- The curved part, which is one-fourth of the circumference of the circle.

3. **Calculate the Length of the Curved Part**:
- The circumference of a full circle is given by the formula:
\[ C = 2\pi r \]
- Since we only need one-fourth of the circle (the curved part of the quadrant), we divide the circumference by 4:
\[ \text{Curved Length} = \frac{2\pi r}{4} = \frac{\pi r}{2} \]

4. **Add the Lengths Together**:
- The total perimeter (P) of the quadrant is the sum of the lengths of the two straight sides and the curved part:
\[ P = r + r + \frac{\pi r}{2} \]
- Simplifying this gives:
\[ P = 2r + \frac{\pi r}{2} \]

5. **Factor Out 'r'**:
- To express the perimeter in a more compact form, we can factor out 'r':
\[ P = r \left(2 + \frac{\pi}{2}\right) \]

### Final Result:
Thus, the perimeter of a quadrant of a circle with radius 'r' is:
\[ P = r \left(2 + \frac{\pi}{2}\right) \]