If the difference between areas of the circumcircle and the incircle of an equllateral triangle is 44 `cm^(2)`, then the area of the triangle is (Take `pi = (22)/(7)`)

Options

A. `28 cm^(2)`
B. `7 sqrt3 cm^(2)`
C. `14 sqrt3 cm^(2)`
D. `21 cm^2`

Answer

To solve the problem, we need to find the area of an equilateral triangle given that the difference between the areas of its circumcircle and incircle is 44 cm². We will follow these steps:

### Step 1: Define the side length of the equilateral triangle
Let the side length of the equilateral triangle be \( a \).

### Step 2: Calculate the radius of the incircle
The radius \( r \) of the incircle of an equilateral triangle is given by the formula:
\[ r = \frac{a}{2\sqrt{3}} \]

### Step 3: Calculate the radius of the circumcircle
The radius \( R \) of the circumcircle of an equilateral triangle is given by the formula:
\[ R = \frac{a}{\sqrt{3}} \]

### Step 4: Calculate the areas of the circumcircle and incircle
The area \( A_{circum} \) of the circumcircle is:
\[ A_{circum} = \pi R^2 = \pi \left(\frac{a}{\sqrt{3}}\right)^2 = \pi \frac{a^2}{3} \]

The area \( A_{in} \) of the incircle is:
\[ A_{in} = \pi r^2 = \pi \left(\frac{a}{2\sqrt{3}}\right)^2 = \pi \frac{a^2}{12} \]

### Step 5: Set up the equation for the difference in areas
According to the problem, the difference between the areas of the circumcircle and the incircle is 44 cm²:
\[ A_{circum} - A_{in} = 44 \]
Substituting the areas we calculated:
\[ \pi \frac{a^2}{3} - \pi \frac{a^2}{12} = 44 \]

### Step 6: Simplify the equation
Factor out \( \pi \):
\[ \pi \left(\frac{a^2}{3} - \frac{a^2}{12}\right) = 44 \]
Finding a common denominator (which is 12):
\[ \frac{4a^2}{12} - \frac{a^2}{12} = \frac{3a^2}{12} \]
So we have:
\[ \pi \frac{3a^2}{12} = 44 \]

### Step 7: Solve for \( a^2 \)
Multiply both sides by \( \frac{12}{3\pi} \):
\[ a^2 = \frac{44 \times 12}{3\pi} \]

### Step 8: Substitute \( \pi = \frac{22}{7} \)
Substituting the value of \( \pi \):
\[ a^2 = \frac{44 \times 12}{3 \times \frac{22}{7}} = \frac{44 \times 12 \times 7}{3 \times 22} \]
Simplifying:
\[ = \frac{44 \times 12 \times 7}{66} = \frac{44 \times 12 \times 7}{66} = \frac{44 \times 12}{6} = 88 \times 2 = 176 \]

### Step 9: Calculate the area of the triangle
The area \( A \) of an equilateral triangle is given by:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Substituting \( a^2 = 176 \):
\[ A = \frac{\sqrt{3}}{4} \times 176 = 44\sqrt{3} \text{ cm}^2 \]

### Final Answer
The area of the triangle is \( 44\sqrt{3} \text{ cm}^2 \).