Find the radius of a circlein which a central angle of `45^(@)` intercepts an arc of length`33 cm`. ( Take `pi= 22/7`.)

Answer

To find the radius of a circle in which a central angle of \(45^\circ\) intercepts an arc of length \(33 \, \text{cm}\), we can follow these steps:

### Step 1: Convert the angle from degrees to radians
To convert degrees to radians, we use the formula:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
For \(45^\circ\):
\[ \text{radians} = 45 \times \frac{\pi}{180} = \frac{\pi}{4} \, \text{radians} \]

**Hint for Step 1:** Remember that \(180^\circ\) is equivalent to \(\pi\) radians.

### Step 2: Use the formula for the length of an arc
The formula for the length of an arc \(L\) is given by:
\[ L = \theta \times r \]
where \(L\) is the length of the arc, \(\theta\) is the angle in radians, and \(r\) is the radius of the circle.

### Step 3: Rearrange the formula to solve for the radius
We need to find the radius \(r\), so we rearrange the formula:
\[ r = \frac{L}{\theta} \]

### Step 4: Substitute the known values
We know that the length of the arc \(L = 33 \, \text{cm}\) and \(\theta = \frac{\pi}{4}\):
\[ r = \frac{33}{\frac{\pi}{4}} = 33 \times \frac{4}{\pi} \]

### Step 5: Substitute \(\pi\) with \(\frac{22}{7}\)
Now, substitute \(\pi\) with \(\frac{22}{7}\):
\[ r = 33 \times \frac{4}{\frac{22}{7}} = 33 \times \frac{4 \times 7}{22} \]

### Step 6: Simplify the expression
Now, simplify the expression:
\[ r = 33 \times \frac{28}{22} = 33 \times \frac{14}{11} \]
Calculating this gives:
\[ r = 33 \times \frac{14}{11} = \frac{462}{11} = 42 \, \text{cm} \]

### Final Answer
Thus, the radius of the circle is:
\[ \boxed{42 \, \text{cm}} \]

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