The area of an equilateral triangle inscribed in a circle of radius `4 cm`, is

Options

A. `12cm^(2)`
B. `9sqrt(3) cm^(2)`
C. `8sqrt(3)cm^(2)`
D. `12sqrt(3)cm^(2)`

Answer

To find the area of an equilateral triangle inscribed in a circle of radius 4 cm, we can follow these steps:

### Step 1: Understand the relationship between the radius and the side of the triangle.
The radius \( R \) of the circumcircle of an equilateral triangle is related to the side length \( a \) of the triangle by the formula:
\[ R = \frac{a}{\sqrt{3}} \]
Given that the radius \( R = 4 \) cm, we can rearrange the formula to find \( a \):
\[ a = R \cdot \sqrt{3} = 4 \cdot \sqrt{3} \]

### Step 2: Calculate the side length of the triangle.
Substituting the value of \( R \):
\[ a = 4 \sqrt{3} \text{ cm} \]

### Step 3: Calculate the height of the triangle.
The height \( h \) of an equilateral triangle can be calculated using the formula:
\[ h = \frac{\sqrt{3}}{2} a \]
Substituting the value of \( a \):
\[ h = \frac{\sqrt{3}}{2} \cdot (4 \sqrt{3}) = \frac{4 \cdot 3}{2} = 6 \text{ cm} \]

### Step 4: Calculate the area of the triangle.
The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is equal to the side length \( a \):
\[ A = \frac{1}{2} \times (4 \sqrt{3}) \times 6 \]
Calculating this gives:
\[ A = \frac{1}{2} \times 24 \sqrt{3} = 12 \sqrt{3} \text{ cm}^2 \]

### Final Answer:
The area of the equilateral triangle inscribed in a circle of radius 4 cm is:
\[ \boxed{12 \sqrt{3} \text{ cm}^2} \]

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