What is the ratio of radius of incircle and circumcircle for an equilateral triangle with an area of `22 sqrt""3` square units?
A. 1/2
B. 1/4
C. 1/3
D. 2/3

Options

A. C
B. B
C. A
D. D

Answer

To find the ratio of the radius of the incircle to the circumcircle for an equilateral triangle with an area of \(22 \sqrt{3}\) square units, we can follow these steps:

### Step 1: Use the formula for the area of an equilateral triangle
The area \(A\) of an equilateral triangle with side length \(a\) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Given that the area is \(22 \sqrt{3}\), we can set up the equation:
\[ \frac{\sqrt{3}}{4} a^2 = 22 \sqrt{3} \]

### Step 2: Solve for \(a^2\)
To eliminate \(\sqrt{3}\) from both sides, we can divide both sides by \(\sqrt{3}\):
\[ \frac{1}{4} a^2 = 22 \]
Now, multiply both sides by 4:
\[ a^2 = 88 \]

### Step 3: Find the side length \(a\)
Taking the square root of both sides gives us:
\[ a = \sqrt{88} = 2\sqrt{22} \]

### Step 4: Calculate the circumradius \(R\)
The circumradius \(R\) of an equilateral triangle is given by:
\[ R = \frac{a}{\sqrt{3}} \]
Substituting the value of \(a\):
\[ R = \frac{2\sqrt{22}}{\sqrt{3}} = \frac{2\sqrt{22}}{\sqrt{3}} \]

### Step 5: Calculate the inradius \(r\)
The inradius \(r\) of an equilateral triangle is given by:
\[ r = \frac{a}{2\sqrt{3}} \]
Substituting the value of \(a\):
\[ r = \frac{2\sqrt{22}}{2\sqrt{3}} = \frac{\sqrt{22}}{\sqrt{3}} \]

### Step 6: Find the ratio of inradius to circumradius
Now we can find the ratio of the inradius \(r\) to the circumradius \(R\):
\[ \text{Ratio} = \frac{r}{R} = \frac{\frac{\sqrt{22}}{\sqrt{3}}}{\frac{2\sqrt{22}}{\sqrt{3}}} \]
This simplifies to:
\[ \text{Ratio} = \frac{\sqrt{22}}{2\sqrt{22}} = \frac{1}{2} \]

### Conclusion
Thus, the ratio of the radius of the incircle to the circumcircle for the given equilateral triangle is:
\[ \frac{1}{2} \]

### Answer
The correct option is **A. \( \frac{1}{2} \)**.