The area of the incircle of an equilateral triangle of side 42 cm is (Take `pi = (22)/(7)`):

Options

A. `231 cm^(2)`
B. `462 cm^(2)`
C. `22sqrt(3) cm^(2)`
D. `924 cm^(2)`

Answer

To find the area of the incircle of an equilateral triangle with a side length of 42 cm, we can follow these steps:

### Step 1: Find the radius of the incircle

The formula for the radius (r) of the incircle of an equilateral triangle is given by:
\[ r = \frac{s \sqrt{3}}{6} \]
where \(s\) is the side length of the triangle.

Given that the side length \(s = 42\) cm, we can substitute this value into the formula.

\[ r = \frac{42 \sqrt{3}}{6} \]

### Step 2: Simplify the expression for the radius

Now, simplify the expression:
\[ r = \frac{42}{6} \sqrt{3} = 7 \sqrt{3} \text{ cm} \]

### Step 3: Calculate the area of the incircle

The area (A) of the incircle can be calculated using the formula:
\[ A = \pi r^2 \]

Substituting the value of \(r\) into the area formula:
\[ A = \pi (7 \sqrt{3})^2 \]

### Step 4: Simplify the area expression

Now, calculate \( (7 \sqrt{3})^2 \):
\[ (7 \sqrt{3})^2 = 49 \times 3 = 147 \]

So, the area becomes:
\[ A = \pi \times 147 \]

### Step 5: Substitute the value of \(\pi\)

Now, substitute \(\pi = \frac{22}{7}\):
\[ A = \frac{22}{7} \times 147 \]

### Step 6: Calculate the area

To simplify:
\[ A = 22 \times 21 = 462 \text{ cm}^2 \]

### Final Answer

The area of the incircle of the equilateral triangle is \(462 \text{ cm}^2\).

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