A square, a circle and equilateral triangle have same perimeter. Consider the following statements. `I`. The area of square is greater than the area of the triangle. `II`. The area of circle is less than the area of triangle. Which of the statement `is//are` correct?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To solve the problem, we need to analyze the areas of a square, a circle, and an equilateral triangle, all of which have the same perimeter. We will evaluate the two statements provided.

### Step-by-Step Solution:

1. **Define the Perimeter**:
Let the perimeter \( P \) be the same for all three shapes.

2. **Perimeter of the Square**:
The perimeter of a square with side length \( a \) is given by:
\[ P = 4a \implies a = \frac{P}{4} \]

3. **Perimeter of the Circle**:
The perimeter (circumference) of a circle with radius \( r \) is given by:
\[ P = 2\pi r \implies r = \frac{P}{2\pi} \]

4. **Perimeter of the Equilateral Triangle**:
The perimeter of an equilateral triangle with side length \( b \) is given by:
\[ P = 3b \implies b = \frac{P}{3} \]

5. **Calculate the Areas**:
- **Area of the Square**:
\[ \text{Area}_{\text{square}} = a^2 = \left(\frac{P}{4}\right)^2 = \frac{P^2}{16} \]

- **Area of the Circle**:
\[ \text{Area}_{\text{circle}} = \pi r^2 = \pi \left(\frac{P}{2\pi}\right)^2 = \frac{P^2}{4\pi} \]

- **Area of the Equilateral Triangle**:
The area of an equilateral triangle is given by:
\[ \text{Area}_{\text{triangle}} = \frac{\sqrt{3}}{4} b^2 = \frac{\sqrt{3}}{4} \left(\frac{P}{3}\right)^2 = \frac{\sqrt{3} P^2}{36} \]

6. **Compare Areas**:
- **Statement I**: The area of the square is greater than the area of the triangle.
\[ \frac{P^2}{16} \text{ vs } \frac{\sqrt{3} P^2}{36} \]
To compare, we can simplify:
\[ \frac{P^2}{16} \cdot \frac{36}{P^2} \text{ vs } \sqrt{3} \implies \frac{36}{16} \text{ vs } \sqrt{3} \implies 2.25 \text{ vs } 1.732 \]
Since \( 2.25 > 1.732 \), Statement I is **True**.

- **Statement II**: The area of the circle is less than the area of the triangle.
\[ \frac{P^2}{4\pi} \text{ vs } \frac{\sqrt{3} P^2}{36} \]
To compare:
\[ \frac{P^2}{4\pi} \cdot \frac{36}{P^2} \text{ vs } \sqrt{3} \implies \frac{36}{4\pi} \text{ vs } \sqrt{3} \implies \frac{9}{\pi} \text{ vs } 1.732 \]
Since \( \frac{9}{3.14} \approx 2.87 > 1.732 \), Statement II is **False**.

### Conclusion:
- Statement I is **True**.
- Statement II is **False**.

Thus, the correct answer is that only Statement I is correct.