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

Options

A. `28 cm^(2)`
B. `7sqrt(3) cm^(2)`
C. `14sqrt(3) cm^(2)`
D. `21 cm^(2)`

Answer

To solve the problem, we need to find the area of an equilateral triangle given the difference between the areas of its circumcircle and incircle is 44 cm².

### Step-by-Step Solution:

1. **Understand the relationship between the circumradius (R) and inradius (r) of an equilateral triangle:**
For an equilateral triangle, the circumradius (R) and inradius (r) are related by the formula:
\[ R = 2r \]
This means the circumradius is twice the inradius.

2. **Calculate the areas of the circumcircle and incircle:**
The area of the circumcircle is given by:
\[ \text{Area of circumcircle} = \pi R^2 \]
The area of the incircle is given by:
\[ \text{Area of incircle} = \pi r^2 \]

3. **Set up the equation for the difference in areas:**
The difference between the areas of the circumcircle and incircle is given as:
\[ \text{Area of circumcircle} - \text{Area of incircle} = 44 \]
Substituting the areas:
\[ \pi R^2 - \pi r^2 = 44 \]
Factoring out \(\pi\):
\[ \pi (R^2 - r^2) = 44 \]

4. **Use the difference of squares:**
Since \(R^2 - r^2\) can be expressed as:
\[ R^2 - r^2 = (R - r)(R + r) \]
And since \(R = 2r\):
\[ R - r = 2r - r = r \]
\[ R + r = 2r + r = 3r \]
Therefore:
\[ R^2 - r^2 = r \cdot 3r = 3r^2 \]

5. **Substitute back into the equation:**
Now we have:
\[ \pi (3r^2) = 44 \]
Solving for \(r^2\):
\[ 3r^2 = \frac{44}{\pi} \]
Substituting \(\pi = \frac{22}{7}\):
\[ 3r^2 = \frac{44 \times 7}{22} = 14 \]
\[ r^2 = \frac{14}{3} \]

6. **Find the circumradius (R):**
Since \(R = 2r\), we can find \(R^2\):
\[ R^2 = (2r)^2 = 4r^2 = 4 \times \frac{14}{3} = \frac{56}{3} \]

7. **Calculate the area of the triangle:**
The area \(A\) of an equilateral triangle can be calculated using the inradius \(r\):
\[ A = \frac{1}{2} \times \text{perimeter} \times r \]
The perimeter of an equilateral triangle with side length \(a\) is \(3a\), and the relationship between \(r\) and \(a\) is:
\[ r = \frac{a \sqrt{3}}{6} \implies a = \frac{6r}{\sqrt{3}} = 2\sqrt{3}r \]
The area can also be expressed as:
\[ A = \frac{\sqrt{3}}{4} a^2 = \frac{\sqrt{3}}{4} (2\sqrt{3}r)^2 = \frac{\sqrt{3}}{4} \cdot 12r^2 = 3\sqrt{3}r^2 \]
Substituting \(r^2 = \frac{14}{3}\):
\[ A = 3\sqrt{3} \cdot \frac{14}{3} = 14\sqrt{3} \]

### Final Answer:
The area of the triangle is \(14\sqrt{3} \, \text{cm}^2\).