The difference between the area of circle and square of radius r is 105 cm^2. What will be the circumference of the circle?

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Write down the formulas for the areas of the circle and the square.
- The area of the circle (A1) is given by the formula:
\[ A1 = \pi r^2 \]
- The area of the square (A2) is given by the formula:
\[ A2 = r^2 \]

### Step 2: Set up the equation based on the given information.
According to the problem, the difference between the area of the circle and the area of the square is 105 cm²:
\[ A1 - A2 = 105 \]
Substituting the formulas from Step 1:
\[ \pi r^2 - r^2 = 105 \]

### Step 3: Simplify the equation.
Factor out \(r^2\) from the left side:
\[ r^2(\pi - 1) = 105 \]

### Step 4: Solve for \(r^2\).
Now, we can isolate \(r^2\):
\[ r^2 = \frac{105}{\pi - 1} \]
Using \(\pi \approx \frac{22}{7}\), we can substitute this value:
\[ r^2 = \frac{105}{\frac{22}{7} - 1} \]
Calculating \(\frac{22}{7} - 1\):
\[ \frac{22}{7} - 1 = \frac{22}{7} - \frac{7}{7} = \frac{15}{7} \]
Now substituting this back:
\[ r^2 = \frac{105}{\frac{15}{7}} = 105 \times \frac{7}{15} \]

### Step 5: Calculate \(r^2\).
Now simplify:
\[ r^2 = \frac{105 \times 7}{15} = \frac{735}{15} = 49 \]

### Step 6: Find \(r\).
Taking the square root of both sides:
\[ r = \sqrt{49} = 7 \text{ cm} \]

### Step 7: Calculate the circumference of the circle.
The formula for the circumference (C) of a circle is:
\[ C = 2\pi r \]
Substituting \(r = 7\) cm and \(\pi \approx \frac{22}{7}\):
\[ C = 2 \times \frac{22}{7} \times 7 \]
The \(7\) in the numerator and denominator cancels out:
\[ C = 2 \times 22 = 44 \text{ cm} \]

### Final Answer:
The circumference of the circle is \(44 \text{ cm}\).

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