To find the area of the square inscribed inside a circle, we need to determine the length of the side of the square.

Let's consider the equilateral triangle. Since all sides of an equilateral triangle are equal, the length of each side of the triangle is 3.

The circle inscribed inside the equilateral triangle touches the midpoint of each side of the triangle. These midpoints will form the vertices of the square.

If we draw a line segment from one of the vertices of the equilateral triangle to the center of the circle, it will be the radius of the circle. Since the radius of the circle is perpendicular to the side of the square, it will also be the diagonal of the square.

Let's call the side of the square "s" and the radius of the circle "r". The diagonal of a square can be found using the formula: diagonal = s√2.

Since the radius of the circle is also the diagonal of the square, we have: r = s√2.

We know that the radius of the circle is the distance from the center of the circle to one of the vertices of the equilateral triangle. By drawing a line segment from the center of the circle to the midpoint of one side of the equilateral triangle, we can create a right-angled triangle.

The hypotenuse of the right-angled triangle is the radius of the circle, which is "r". The base of the right-angled triangle is half the length of one side of the equilateral triangle, which is 3/2. The height of the right-angled triangle is the length from the center of the circle to the midpoint of one side of the equilateral triangle, which can be found using the Pythagorean theorem.

Using the Pythagorean theorem, we have:
r² = (3/2)² + h²,
r² = 9/4 + h².

Since r = s√2, we can substitute r in the equation above:
(s√2)² = 9/4 + h²,
2s² = 9/4 + h².

Since the height of the right-angled triangle is equal to the side of the square, h = s.

Substituting h = s in the equation above, we get:
2s² = 9/4 + s²,
s² = 9/4,
s = 3/2.

Therefore, the side of the square is 3/2.

The area of a square is given by the formula: area = side².

Substituting the value of the side of the square, we have:
area = (3/2)² = 9/4.

Therefore, the area of the square inscribed inside the circle, which is in turn inscribed inside the equilateral triangle of side 3, is 9/4 square units.