The length of each side of an equilateral triagle is `14sqrt(3)cm`. The area of the incircle (in `cm^(2)`), is

Options

A. 450
B. 308
C. 154
D. 77

Answer

To find the area of the incircle of an equilateral triangle with a side length of \( 14\sqrt{3} \, \text{cm} \), we will follow these steps:

### Step 1: Identify the formula for the radius of the incircle
The radius \( r \) of the incircle of an equilateral triangle can be calculated using the formula:
\[ r = \frac{A}{s} \]
where \( A \) is the area of the triangle and \( s \) is the semi-perimeter.

### Step 2: Calculate the semi-perimeter
For an equilateral triangle, the semi-perimeter \( s \) is given by:
\[ s = \frac{3a}{2} \]
where \( a \) is the length of a side. Here, \( a = 14\sqrt{3} \, \text{cm} \).
\[ s = \frac{3 \times 14\sqrt{3}}{2} = 21\sqrt{3} \, \text{cm} \]

### Step 3: Calculate the area of the triangle
The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Substituting \( a = 14\sqrt{3} \):
\[ A = \frac{\sqrt{3}}{4} (14\sqrt{3})^2 = \frac{\sqrt{3}}{4} \times 588 = 147\sqrt{3} \, \text{cm}^2 \]

### Step 4: Calculate the radius of the incircle
Now, substitute the values of \( A \) and \( s \) into the formula for \( r \):
\[ r = \frac{A}{s} = \frac{147\sqrt{3}}{21\sqrt{3}} = 7 \, \text{cm} \]

### Step 5: Calculate the area of the incircle
The area \( A_{circle} \) of the incircle can be calculated using the formula:
\[ A_{circle} = \pi r^2 \]
Substituting \( r = 7 \, \text{cm} \):
\[ A_{circle} = \pi (7)^2 = 49\pi \, \text{cm}^2 \]
Using \( \pi \approx \frac{22}{7} \):
\[ A_{circle} \approx 49 \times \frac{22}{7} = 154 \, \text{cm}^2 \]

### Final Answer
The area of the incircle is \( 154 \, \text{cm}^2 \).

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