An equilateral triangle is inscribed in a circle. If a side of the triangle is 12 cm. If a side of the triangle is 12 cm. Find the area of the circle.

Answer

To find the area of the circle in which an equilateral triangle with a side length of 12 cm is inscribed, we can follow these steps:

### Step 1: Understand the relationship between the triangle and the circle
An equilateral triangle inscribed in a circle means that all three vertices of the triangle touch the circumference of the circle. The center of the circle is also the centroid of the triangle.

### Step 2: Determine the radius of the circle
For an equilateral triangle with side length \( a \), the radius \( R \) of the circumscribed circle (circumcircle) can be calculated using the formula:
\[ R = \frac{a}{\sqrt{3}} \]
Given that the side length \( a = 12 \) cm, we can substitute this value into the formula:
\[ R = \frac{12}{\sqrt{3}} = \frac{12 \sqrt{3}}{3} = 4 \sqrt{3} \text{ cm} \]

### Step 3: Calculate the area of the circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi R^2 \]
Substituting the value of \( R \):
\[ A = \pi (4 \sqrt{3})^2 \]
Calculating \( (4 \sqrt{3})^2 \):
\[ (4 \sqrt{3})^2 = 16 \times 3 = 48 \]
Thus, the area becomes:
\[ A = 48 \pi \text{ cm}^2 \]

### Final Answer
The area of the circle is \( 48 \pi \) cm².

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