If the perimeter of a triangle is 6, then its maximum area is

Options

A. `sqrt(3)`
B. 3
C. 4
D. 2

Answer

To find the maximum area of a triangle with a given perimeter of 6 units, we can follow these steps:

### Step 1: Understand the relationship between the perimeter and the sides of the triangle.
The perimeter \( P \) of a triangle is the sum of the lengths of its sides. For an equilateral triangle, all sides are equal. If we denote the length of each side as \( a \), then the perimeter can be expressed as:
\[ P = 3a \]
Given that the perimeter is 6, we can set up the equation:
\[ 3a = 6 \]

### Step 2: Solve for the length of each side.
To find the length of each side \( a \), we can rearrange the equation:
\[ a = \frac{6}{3} = 2 \]

### Step 3: Use the formula for the area of an equilateral triangle.
The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Now, substituting the value of \( a \) we found:
\[ A = \frac{\sqrt{3}}{4} (2)^2 \]

### Step 4: Calculate the area.
Now we can simplify the expression:
\[ A = \frac{\sqrt{3}}{4} \cdot 4 = \sqrt{3} \]

### Conclusion:
Thus, the maximum area of the triangle with a perimeter of 6 units is:
\[ \boxed{\sqrt{3}} \]

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