Find the perimeter of a quadrant of a circle of radius 'r'.

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 of a circle is one-fourth of the circle. It consists of two straight sides (the radii) and one curved side (the arc).

### Step 2: Identify the Components of the Perimeter
The perimeter (P) of the quadrant can be expressed as the sum of:
- The length of the two straight sides (the radii)
- The length of the arc

### Step 3: Calculate the Length of the Straight Sides
Since both sides OA and OB are the radii of the circle:
- OA = r
- OB = r

Thus, the total length of the straight sides is:
\[ OA + OB = r + r = 2r \]

### Step 4: Calculate the Length of the Arc
The circumference (C) of a full circle is given by the formula:
\[ C = 2\pi r \]
Since we only need one-fourth of the circumference for the quadrant, we calculate:
\[ \text{Arc length (AB)} = \frac{1}{4} \times C = \frac{1}{4} \times 2\pi r = \frac{\pi r}{2} \]

### Step 5: Combine the Lengths to Find the Perimeter
Now, we can combine the lengths of the straight sides and the arc to find the total perimeter of the quadrant:
\[ P = OA + OB + \text{Arc length} \]
\[ P = 2r + \frac{\pi r}{2} \]

### Step 6: Factor Out 'r'
To simplify, we can factor 'r' out of the expression:
\[ P = r \left(2 + \frac{\pi}{2}\right) \]

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