Circumference of two circles are equal. Is it necessary that their areas be equal ? Why ?

Explanation

To determine whether the areas of two circles are equal when their circumferences are equal, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Circumference**:
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.

2. **Set Up the Problem**:
Let’s consider two circles with radii \( r_1 \) and \( r_2 \). According to the problem, the circumferences of these two circles are equal:
\[ 2\pi r_1 = 2\pi r_2 \]

3. **Simplify the Equation**:
We can simplify the equation by dividing both sides by \( 2\pi \):
\[ r_1 = r_2 \]
This means that the radii of both circles are equal.

4. **Understand the Formula for Area**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]

5. **Calculate the Areas of Both Circles**:
- The area of the first circle is:
\[ A_1 = \pi r_1^2 \]
- The area of the second circle is:
\[ A_2 = \pi r_2^2 \]

6. **Substitute the Equal Radii**:
Since we established that \( r_1 = r_2 \), we can substitute \( r_2 \) with \( r_1 \):
\[ A_2 = \pi r_1^2 \]

7. **Compare the Areas**:
Now we can see that:
\[ A_1 = \pi r_1^2 \quad \text{and} \quad A_2 = \pi r_1^2 \]
Therefore, \( A_1 = A_2 \).

8. **Conclusion**:
Since the areas of both circles are equal, we conclude that if the circumferences of two circles are equal, their areas must also be equal.

### Final Answer:
Yes, it is necessary that their areas be equal because equal circumferences imply equal radii, and equal radii lead to equal areas.

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