The perimeters of a circle, a square and an equilateral triangle are same and their areas are C,S and T respectively. Which of the following statement is true ?

Options

A. C = S = T
B. C > S >T
C. C < S < T
D. S < C < T

Answer

To solve the problem, we need to analyze the areas of a circle, a square, and an equilateral triangle given that their perimeters are the same. Let's denote the following:

- Let \( R \) be the radius of the circle.
- Let \( a \) be the side length of the square.
- Let \( b \) be the side length of the equilateral triangle.

### Step 1: Set up the perimeter equations

The perimeters of the three shapes are given by:
- Circle: \( P_{circle} = 2\pi R \)
- Square: \( P_{square} = 4a \)
- Equilateral Triangle: \( P_{triangle} = 3b \)

Since the perimeters are equal, we can set up the following equations:
\[ 2\pi R = 4a = 3b \]

### Step 2: Express \( a \) and \( b \) in terms of \( R \)

From the perimeter equations, we can express \( a \) and \( b \) in terms of \( R \):
1. From \( 2\pi R = 4a \):
\[ a = \frac{2\pi R}{4} = \frac{\pi R}{2} \]
2. From \( 2\pi R = 3b \):
\[ b = \frac{2\pi R}{3} \]

### Step 3: Calculate the areas of the shapes

Now we can calculate the areas of each shape:
1. Area of the circle:
\[ A_{circle} = \pi R^2 \]

2. Area of the square:
\[ A_{square} = a^2 = \left(\frac{\pi R}{2}\right)^2 = \frac{\pi^2 R^2}{4} \]

3. Area of the equilateral triangle:
The area of an equilateral triangle is given by:
\[ A_{triangle} = \frac{\sqrt{3}}{4} b^2 = \frac{\sqrt{3}}{4} \left(\frac{2\pi R}{3}\right)^2 = \frac{\sqrt{3}}{4} \cdot \frac{4\pi^2 R^2}{9} = \frac{\sqrt{3}\pi^2 R^2}{9} \]

### Step 4: Compare the areas

Now we have the areas:
- \( A_{circle} = \pi R^2 \)
- \( A_{square} = \frac{\pi^2 R^2}{4} \)
- \( A_{triangle} = \frac{\sqrt{3}\pi^2 R^2}{9} \)

To compare these areas, we can look at their ratios:
1. \( A_{circle} : A_{square} : A_{triangle} \)
\[ = \pi R^2 : \frac{\pi^2 R^2}{4} : \frac{\sqrt{3}\pi^2 R^2}{9} \]

Dividing each term by \( R^2 \):
\[ = \pi : \frac{\pi^2}{4} : \frac{\sqrt{3}\pi^2}{9} \]

### Step 5: Simplify the ratios

To simplify, we can multiply through by \( 36 \) (the least common multiple of the denominators):
\[ = 36\pi : 9\pi^2 : 4\sqrt{3}\pi^2 \]

This gives us:
\[ = 36 : 9\pi : 4\sqrt{3}\pi \]

### Conclusion

From the above ratios, we can determine the order of the areas:
- The area of the circle is the largest, followed by the area of the square, and then the area of the equilateral triangle.

Thus, the correct statement is:
**The area of the circle is greater than the area of the square, and the area of the square is greater than the area of the equilateral triangle.**