The area of a circle inscribed in an equilateral triangle is 462 cm.The perimeter of the triangle is -----

Options

A. A)`42sqrt3` cms
B. B)126 cms
C. C)72.6 cms
D. D)168 cms

Answer

To solve the problem, we need to find the perimeter of an equilateral triangle given the area of the inscribed circle (incircle). Here are the steps to find the perimeter:

### Step 1: Use the area of the circle to find the radius
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Given that the area of the circle is \( 462 \, \text{cm}^2 \), we can set up the equation:
\[ \pi r^2 = 462 \]

### Step 2: Solve for \( r^2 \)
Using \( \pi \approx \frac{22}{7} \), we can rewrite the equation:
\[ \frac{22}{7} r^2 = 462 \]
To isolate \( r^2 \), multiply both sides by \( \frac{7}{22} \):
\[ r^2 = 462 \times \frac{7}{22} \]

### Step 3: Simplify the right side
Calculating the right side:
\[ r^2 = 462 \times \frac{7}{22} = 462 \div 22 \times 7 = 21 \times 7 = 147 \]

### Step 4: Find the radius \( r \)
Now, take the square root to find \( r \):
\[ r = \sqrt{147} = \sqrt{49 \times 3} = 7\sqrt{3} \, \text{cm} \]

### Step 5: Relate the radius to the side length of the triangle
For an equilateral triangle, the radius \( r \) of the incircle is related to the side length \( a \) by the formula:
\[ r = \frac{a \sqrt{3}}{6} \]
Substituting \( r = 7\sqrt{3} \):
\[ 7\sqrt{3} = \frac{a \sqrt{3}}{6} \]

### Step 6: Solve for \( a \)
To find \( a \), multiply both sides by \( 6 \):
\[ 42\sqrt{3} = a\sqrt{3} \]
Now, divide both sides by \( \sqrt{3} \):
\[ a = 42 \, \text{cm} \]

### Step 7: Calculate the perimeter of the triangle
The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3a \]
Substituting \( a = 42 \):
\[ P = 3 \times 42 = 126 \, \text{cm} \]

### Final Answer
The perimeter of the triangle is \( 126 \, \text{cm} \).

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