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

Answer

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

### Step 1: Define the Circumference of the Circles
Let the circumference of the first circle be \( C_1 \) and the circumference of the second circle be \( C_2 \). According to the problem, we have:
\[ C_1 = C_2 \]

### Step 2: Write the Formula for Circumference
The formula for the circumference of a circle is given by:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle. Therefore, we can express the circumferences of the two circles as:
\[ C_1 = 2\pi r_1 \quad \text{and} \quad C_2 = 2\pi r_2 \]

### Step 3: Set the Circumferences Equal
Since \( C_1 = C_2 \), we can write:
\[ 2\pi r_1 = 2\pi r_2 \]

### Step 4: Simplify the Equation
We can divide both sides of the equation by \( 2\pi \) (assuming \( \pi \neq 0 \)):
\[ r_1 = r_2 \]

### Step 5: Write the Area 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 6: Substitute the Radius
Since we have established that \( r_1 = r_2 \), we can substitute \( r_2 \) for \( r_1 \) in the area formulas:
\[ A_1 = \pi r_1^2 \quad \text{and} \quad A_2 = \pi r_1^2 \]

### Step 7: Conclude the Areas are Equal
From the above equations, we can 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.

### Summary
We have demonstrated that if \( C_1 = C_2 \), then \( A_1 = A_2 \).

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