Areas of two circles are equal. Is it necessary that their circumferences are equal ? Why ?

Explanation

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

### Step-by-Step Solution:

1. **Understand the Area of a Circle**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

2. **Set Up the Equation for Two Circles**:
Let the radius of the first circle be \( r_1 \) and the radius of the second circle be \( r_2 \). According to the problem, the areas of the two circles are equal:
\[ \pi r_1^2 = \pi r_2^2 \]

3. **Simplify the Equation**:
We can cancel \( \pi \) from both sides of the equation:
\[ r_1^2 = r_2^2 \]

4. **Take the Square Root**:
Taking the square root of both sides gives:
\[ r_1 = r_2 \]
This shows that the radii of the two circles are equal.

5. **Circumference of a Circle**:
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]

6. **Set Up the Circumference for Both Circles**:
The circumference of the first circle is:
\[ C_1 = 2\pi r_1 \]
and the circumference of the second circle is:
\[ C_2 = 2\pi r_2 \]

7. **Substitute the Radii**:
Since we found that \( r_1 = r_2 \), we can substitute \( r_2 \) with \( r_1 \) in the circumference formula:
\[ C_2 = 2\pi r_1 \]

8. **Compare the Circumferences**:
Now we can compare the two circumferences:
\[ C_1 = 2\pi r_1 \quad \text{and} \quad C_2 = 2\pi r_1 \]
This implies:
\[ C_1 = C_2 \]

### Conclusion:
Thus, if the areas of two circles are equal, it is indeed necessary that their circumferences are also equal.

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