An equilateral triangle with a perimeter of 12 is inscribed in a circle. What is the area of circle?

Options

A. `(16pi)/(9)`
B. `(4pisqrt(3))/(3)`
C. `3pi`
D. `(16pi)/(3)`

Answer

To find the area of the circle inscribed around an equilateral triangle with a perimeter of 12, follow these steps:

### Step 1: Determine the side length of the equilateral triangle.
The perimeter of the equilateral triangle is given as 12. Since an equilateral triangle has three equal sides, we can denote the side length as \( x \).

\[ \text{Perimeter} = 3x = 12 \]

To find \( x \):

\[ x = \frac{12}{3} = 4 \]

### Step 2: Understand the relationship between the triangle and the circle.
The radius \( r \) of the circumcircle of an equilateral triangle can be found using the formula:

\[ r = \frac{x}{\sqrt{3}} \]

where \( x \) is the side length of the triangle.

### Step 3: Substitute the side length into the radius formula.
Now substituting \( x = 4 \):

\[ r = \frac{4}{\sqrt{3}} = \frac{4\sqrt{3}}{3} \]

### Step 4: Calculate the area of the circle.
The area \( A \) of a circle is given by the formula:

\[ A = \pi r^2 \]

Substituting \( r = \frac{4\sqrt{3}}{3} \):

\[ A = \pi \left(\frac{4\sqrt{3}}{3}\right)^2 \]

Calculating \( r^2 \):

\[ r^2 = \left(\frac{4\sqrt{3}}{3}\right)^2 = \frac{16 \cdot 3}{9} = \frac{48}{9} = \frac{16}{3} \]

Now substituting back into the area formula:

\[ A = \pi \cdot \frac{16}{3} = \frac{16\pi}{3} \]

### Final Answer:
The area of the circle is

\[ \frac{16\pi}{3} \]