The outer and inner diameters of a circular ring are 34 cm and 32 cm respectively. The area of the ring is :

Options

A. 66 `pi cm^(2)`
B. 60 `pi cm^(2)`
C. 33 `pi cm^(2)`
D. 29 `pi cm^(2)`

Answer

To find the area of the circular ring, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the diameters**:
- The outer diameter (D1) is 34 cm.
- The inner diameter (D2) is 32 cm.

2. **Calculate the radii**:
- The radius of the outer circle (R1) is half of the outer diameter:
\[ R1 = \frac{D1}{2} = \frac{34}{2} = 17 \text{ cm} \]
- The radius of the inner circle (R2) is half of the inner diameter:
\[ R2 = \frac{D2}{2} = \frac{32}{2} = 16 \text{ cm} \]

3. **Use the area formula for circles**:
- The area of the outer circle (A1) is given by:
\[ A1 = \pi R1^2 = \pi (17)^2 = \pi \times 289 \]
- The area of the inner circle (A2) is given by:
\[ A2 = \pi R2^2 = \pi (16)^2 = \pi \times 256 \]

4. **Calculate the area of the ring**:
- The area of the ring (A_ring) is the difference between the area of the outer circle and the area of the inner circle:
\[ A_{ring} = A1 - A2 = \pi \times 289 - \pi \times 256 \]
- Factor out \(\pi\):
\[ A_{ring} = \pi (289 - 256) \]
- Calculate the difference:
\[ 289 - 256 = 33 \]
- Therefore, the area of the ring is:
\[ A_{ring} = 33\pi \text{ cm}^2 \]

### Final Answer:
The area of the ring is \(33\pi \text{ cm}^2\).

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