If an area enclosed by a circle or a square or an equilateral triangle is the same, then the maximum perimeter is possessed by :

Options

A. circle
B. square
C. equilateral triangle
D. triangle and square have equal perimeters greater than that of circle.

Answer

To determine which shape (circle, square, or equilateral triangle) has the maximum perimeter when they all enclose the same area, we can follow these steps:

### Step 1: Define the common area
Let the common area enclosed by the shapes be \( A \) square units.

### Step 2: Calculate the radius of the circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
From this, we can express the radius \( r \) in terms of the area:
\[ r = \sqrt{\frac{A}{\pi}} \]

### Step 3: Calculate the perimeter of the circle
The perimeter (circumference) \( C \) of a circle is given by:
\[ C = 2\pi r \]
Substituting the value of \( r \):
\[ C = 2\pi \sqrt{\frac{A}{\pi}} = 2\sqrt{A\pi} \]

### Step 4: Calculate the side length of the square
For a square with area \( A \), the side length \( s \) is:
\[ s = \sqrt{A} \]

### Step 5: Calculate the perimeter of the square
The perimeter \( P \) of the square is:
\[ P = 4s = 4\sqrt{A} \]

### Step 6: Calculate the side length of the equilateral triangle
The area \( A \) of an equilateral triangle with side length \( a \) is given by:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
From this, we can express the side length \( a \):
\[ a = \sqrt{\frac{4A}{\sqrt{3}}} \]

### Step 7: Calculate the perimeter of the equilateral triangle
The perimeter \( P_t \) of the equilateral triangle is:
\[ P_t = 3a = 3\sqrt{\frac{4A}{\sqrt{3}}} = 3\cdot 2\sqrt{\frac{A}{\sqrt{3}}} = 6\sqrt{\frac{A}{\sqrt{3}}} \]

### Step 8: Compare the perimeters
Now we have the perimeters for each shape:
- Circle: \( 2\sqrt{A\pi} \)
- Square: \( 4\sqrt{A} \)
- Equilateral Triangle: \( 6\sqrt{\frac{A}{\sqrt{3}}} \)

To find which is maximum, we can evaluate these expressions for a specific value of \( A \) or compare their coefficients.

### Conclusion
After evaluating the expressions, we find that the perimeter of the equilateral triangle is the largest among the three shapes when they enclose the same area.

Thus, the maximum perimeter is possessed by the **equilateral triangle**.

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