A circle is inscribed in an equilateral triangle of side a. What is the area of any square inscribed in the circle ?

Options

A. `(a^(2))/(3)`
B. `(a^(2))/(4)`
C. `(a^(2))/(6)`
D. `(a^(2))/(8)`

Answer

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

### Step 1: Find the radius of the inscribed circle (incircle) of the equilateral triangle.
The radius \( R \) of the incircle of an equilateral triangle can be calculated using the formula:
\[ R = \frac{a}{2\sqrt{3}} \]
where \( a \) is the length of a side of the triangle.

### Step 2: Understand the relationship between the square and the circle.
When a square is inscribed in a circle, the diagonal of the square is equal to the diameter of the circle. Therefore, if the radius of the circle is \( R \), the diameter \( D \) of the circle is:
\[ D = 2R \]

### Step 3: Express the side length of the square in terms of the radius.
Let the side length of the square be \( s \). The relationship between the side length \( s \) and the diagonal \( D \) of the square is given by:
\[ D = s\sqrt{2} \]
Thus, we can express \( s \) as:
\[ s = \frac{D}{\sqrt{2}} = \frac{2R}{\sqrt{2}} = R\sqrt{2} \]

### Step 4: Substitute the radius into the side length formula.
Substituting the expression for \( R \):
\[ s = \left(\frac{a}{2\sqrt{3}}\right)\sqrt{2} = \frac{a\sqrt{2}}{2\sqrt{3}} = \frac{a\sqrt{6}}{6} \]

### Step 5: Calculate the area of the square.
The area \( A \) of the square is given by:
\[ A = s^2 = \left(\frac{a\sqrt{6}}{6}\right)^2 = \frac{6a^2}{36} = \frac{a^2}{6} \]

### Final Answer:
The area of the square inscribed in the circle is:
\[ \boxed{\frac{a^2}{6}} \]