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

Options

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

Answer

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

### Step 1: Understand the Quadrant
A quadrant is one-fourth of a circle. When we talk about the perimeter of a quadrant, we need to consider both the curved part and the two straight sides.

### Step 2: Identify the Components of the Perimeter
The perimeter of the quadrant consists of:
1. The curved arc (which is one-fourth of the circumference of the circle).
2. The two straight sides (which are the radii of the circle).

### Step 3: Calculate 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 this for the quadrant, we calculate:
\[ \text{Curved part} = \frac{1}{4} \times 2\pi r = \frac{\pi r}{2} \]

### Step 4: Calculate the Straight Parts
The two straight sides are both equal to the radius 'r'. Therefore, the total length of the straight sides is:
\[ \text{Straight parts} = r + r = 2r \]

### Step 5: Combine the Parts to Find the Total Perimeter
Now, we can combine the lengths of the curved part and the straight parts to find the total perimeter of the quadrant:
\[ \text{Total Perimeter} = \text{Curved part} + \text{Straight parts} = \frac{\pi r}{2} + 2r \]

### Step 6: Simplify the Expression
To express this in a more standard form, we can factor out 'r':
\[ \text{Total Perimeter} = r \left(\frac{\pi}{2} + 2\right) \]

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

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