The area of an equilateral triangle is numerically equal to its perimeter. Find its perimeter correct to 2 decimal places.

Answer

To find the perimeter of an equilateral triangle whose area is numerically equal to its perimeter, we can follow these steps:

### Step 1: Understand the formulas
The area \( A \) of an equilateral triangle with side length \( a \) is given by:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3a \]

### Step 2: Set up the equation
According to the problem, the area is numerically equal to the perimeter:
\[ \frac{\sqrt{3}}{4} a^2 = 3a \]

### Step 3: Simplify the equation
To simplify the equation, we can first eliminate \( a \) from both sides (assuming \( a \neq 0 \)):
\[ \frac{\sqrt{3}}{4} a = 3 \]

### Step 4: Solve for \( a \)
Now, multiply both sides by 4 to get rid of the fraction:
\[ \sqrt{3} a = 12 \]
Now, divide both sides by \( \sqrt{3} \):
\[ a = \frac{12}{\sqrt{3}} = 12 \cdot \frac{\sqrt{3}}{3} = 4\sqrt{3} \]

### Step 5: Calculate the perimeter
Now that we have the side length \( a \), we can find the perimeter:
\[ P = 3a = 3 \cdot 4\sqrt{3} = 12\sqrt{3} \]

### Step 6: Approximate the value
To find the numerical value of \( 12\sqrt{3} \), we use the approximate value of \( \sqrt{3} \approx 1.732 \):
\[ P \approx 12 \cdot 1.732 = 20.784 \]

### Step 7: Round to two decimal places
Finally, rounding \( 20.784 \) to two decimal places gives:
\[ P \approx 20.78 \]

### Final Answer
The perimeter of the equilateral triangle is approximately \( 20.78 \) units.

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