If the circumference of a circle is equal to the perimeter of square, then which one of the following is correct?

Question

Accepted Answer

To determine why Option 2 is correct, we need to analyze the relationship between the circumference of a circle and the perimeter of a square, and how these relate to their respective areas.

### Explanation for Option 2: Area of circle > Area of square

1. **Given Condition**: The circumference of the circle is equal to the perimeter of the square.
   - Let the radius of the circle be $ r $.
   - The circumference of the circle is $ 2\pi r $.
   - Let the side of the square be $ s $.
   - The perimeter of the square is $ 4s $.

According to the problem, $ 2\pi r = 4s $.

2. **Solving for $ s $ in terms of $ r $**:
   $$
   2\pi r = 4s \implies s = \frac{\pi r}{2}
   $$

3. **Calculating Areas**:
   - **Area of the Circle**: 
     $$
     \text{Area of circle} = \pi r^2
     $$
   - **Area of the Square**:
     $$
     \text{Area of square} = s^2 = \left(\frac{\pi r}{2}\right)^2 = \frac{\pi^2 r^2}{4}
     $$

4. **Comparing the Areas**:
   - To compare the areas, we compare $ \pi r^2 $ and $ \frac{\pi^2 r^2}{4} $.
   - Simplifying the comparison:
     $$
     \pi r^2 > \frac{\pi^2 r^2}{4}
     $$
   - Dividing both sides by $ r^2 $ (assuming $ r \neq 0 $):
     $$
     \pi > \frac{\pi^2}{4}
     $$
   - Multiplying both sides by 4:
     $$
     4\pi > \pi^2
     $$
   - Dividing both sides by $ \pi $ (assuming $ \pi \neq 0 $):
     $$
     4 > \pi
     $$
   - Since $ \pi \approx 3.14 $, it is clear that $ 4 > \pi $.

Thus, the area of the circle is greater than the area of the square, confirming that Option 2 is correct.

### Explanation for Why Other Options are Incorrect:

1. **Option 1: Area of circle = Area of square**
   - As shown above, $ \pi r^2 \neq \frac{\pi^2 r^2}{4} $. The inequality $ 4 > \pi $ proves that the area of the circle is greater than the area of the square, not equal.

2. **Option 3: Area of circle < Area of square**
   - The inequality $ 4 > \pi $ directly contradicts this option, as it shows that the area of the circle is greater, not less.

3. **Option 4: None of the above**
   - Since Option 2 is correct, Option 4 is incorrect by default.

In conclusion, the correct answer is Option 2, as the area of the circle is indeed greater than the area of the square when their circumferences and perimeters are equal.