A circle is cirumscribed in an equilateral triangle of side `'l'`. The area of any square inscribed in the circle is :

Options

A. `(4)/(3)l^(2)`
B. `(2)/(3)l^(2)`
C. `(1)/(3)l^(2)`
D. `l^(2)`

Answer

To find the area of a square inscribed in a circle that is circumscribed around an equilateral triangle of side length \( l \), we can follow these steps:

### Step 1: Find the radius of the circumscribed circle
For an equilateral triangle, the radius \( R \) of the circumscribed circle can be calculated using the formula:
\[ R = \frac{l}{\sqrt{3}} \]
This formula arises from the relationship between the side length of the triangle and the radius of the circumscribed circle.

**Hint:** Remember that for an equilateral triangle, the radius of the circumcircle is a function of the side length.

### Step 2: Relate the square's diagonal to the circle's diameter
The diagonal \( d \) of the inscribed square is equal to the diameter of the circumscribed circle. The diameter \( D \) of the circle is given by:
\[ D = 2R = 2 \times \frac{l}{\sqrt{3}} = \frac{2l}{\sqrt{3}} \]

**Hint:** The diagonal of a square inscribed in a circle is equal to the diameter of that circle.

### Step 3: Find the side length of the square
The relationship between the side length \( a \) of the square and its diagonal \( d \) is given by:
\[ d = a\sqrt{2} \]
Setting the diagonal equal to the diameter of the circle, we have:
\[ a\sqrt{2} = \frac{2l}{\sqrt{3}} \]
Solving for \( a \):
\[ a = \frac{2l}{\sqrt{3}\sqrt{2}} = \frac{2l}{\sqrt{6}} = \frac{l\sqrt{2}}{3} \]

**Hint:** Use the relationship between the side length and diagonal of a square to find the side length.

### Step 4: Calculate the area of the square
The area \( A \) of the square is given by:
\[ A = a^2 \]
Substituting the value of \( a \):
\[ A = \left(\frac{l\sqrt{2}}{3}\right)^2 = \frac{2l^2}{9} \]

**Hint:** Remember that the area of a square is the square of its side length.

### Final Answer
Thus, the area of the square inscribed in the circle is:
\[ \boxed{\frac{2l^2}{9}} \]