Show that if the circumferences of two circles are equal, then their areas are also equal.

Answer

To show that if the circumferences of two circles are equal, then their areas are also equal, we can follow these steps:

### Step 1: Define the circumferences of the circles
Let the radius of the first circle be \( R_1 \) and the radius of the second circle be \( R_2 \). The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi R \]
Thus, the circumferences of the two circles can be expressed as:
\[ C_1 = 2\pi R_1 \quad \text{and} \quad C_2 = 2\pi R_2 \]

### Step 2: Set the circumferences equal
According to the problem, the circumferences of the two circles are equal:
\[ C_1 = C_2 \]
This implies:
\[ 2\pi R_1 = 2\pi R_2 \]

### Step 3: Simplify the equation
Since \( 2\pi \) is a constant and can be divided from both sides, we have:
\[ R_1 = R_2 \]

### Step 4: Calculate the areas of the circles
The area \( A \) of a circle is given by the formula:
\[ A = \pi R^2 \]
Thus, the areas of the two circles can be expressed as:
\[ A_1 = \pi R_1^2 \quad \text{and} \quad A_2 = \pi R_2^2 \]

### Step 5: Substitute the equal radii into the area formulas
Since we have established that \( R_1 = R_2 \), we can substitute \( R_2 \) for \( R_1 \) in the area formula:
\[ A_1 = \pi R_1^2 \quad \text{and} \quad A_2 = \pi R_2^2 = \pi R_1^2 \]

### Step 6: Conclude that the areas are equal
From the above, we see that:
\[ A_1 = A_2 \]
Thus, we have shown that if the circumferences of two circles are equal, then their areas are also equal.

### Final Conclusion
Hence, it is proved that if the circumferences of two circles are equal, their areas are also equal.

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