Find the area of a flat circular ring formed by two concentric circles (circles with same centre) whose radii are 9 cm and 5 cm.

Answer

To find the area of a flat circular ring formed by two concentric circles with radii 9 cm and 5 cm, we can follow these steps:

### Step 1: Identify the radii
Let the radius of the outer circle (C1) be \( R_1 = 9 \) cm and the radius of the inner circle (C2) be \( R_2 = 5 \) cm.

### Step 2: Write the formula for the area of the ring
The area of the circular ring can be calculated using the formula:
\[ \text{Area of the ring} = \text{Area of C1} - \text{Area of C2} \]
This can be expressed as:
\[ \text{Area of the ring} = \pi R_1^2 - \pi R_2^2 \]

### Step 3: Factor out π
We can factor out \( \pi \) from the equation:
\[ \text{Area of the ring} = \pi (R_1^2 - R_2^2) \]

### Step 4: Substitute the values of \( R_1 \) and \( R_2 \)
Substituting the values of \( R_1 \) and \( R_2 \):
\[ \text{Area of the ring} = \pi (9^2 - 5^2) \]

### Step 5: Calculate \( R_1^2 \) and \( R_2^2 \)
Calculate the squares:
\[ 9^2 = 81 \quad \text{and} \quad 5^2 = 25 \]
Now substitute these values back into the equation:
\[ \text{Area of the ring} = \pi (81 - 25) \]

### Step 6: Perform the subtraction
Calculate the difference:
\[ 81 - 25 = 56 \]
So, we have:
\[ \text{Area of the ring} = \pi \times 56 \]

### Step 7: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \):
\[ \text{Area of the ring} = \frac{22}{7} \times 56 \]

### Step 8: Perform the multiplication
Calculate \( \frac{22 \times 56}{7} \):
\[ 22 \times 56 = 1232 \]
Now divide by 7:
\[ \frac{1232}{7} = 176 \]

### Step 9: State the final answer
Thus, the area of the flat circular ring is:
\[ \text{Area} = 176 \, \text{cm}^2 \]