To find the area of the square that can be inscribed in a circle of radius 12 cm, we need to understand the relationship between the square and the circle.

Inscribed Square:
An inscribed square is a square that is drawn inside a circle in such a way that all four corners of the square touch the circumference of the circle.

Properties of Inscribed Square:
1. The diagonal of the square is equal to the diameter of the circle.
2. The sides of the square are equal in length.

Approach:
1. We are given the radius of the circle, which is 12 cm.
2. To find the side length of the square, we can use the diagonal-diameter relationship mentioned earlier.
3. The diagonal of the square is equal to the diameter of the circle, which is twice the radius.
Diagonal = 2 * Radius = 2 * 12 cm = 24 cm
4. Since the diagonal of the square is also the hypotenuse of a right-angled triangle formed by the sides of the square, we can use the Pythagorean theorem to find the side length of the square.
(Side)^2 + (Side)^2 = (Diagonal)^2
2 * (Side)^2 = (Diagonal)^2
2 * (Side)^2 = (24 cm)^2
(Side)^2 = (24 cm)^2 / 2
(Side)^2 = 576 cm^2 / 2
(Side)^2 = 288 cm^2
5. Taking the square root of both sides, we find the side length of the square.
Side = √(288 cm^2)
Side ≈ 16.97 cm
6. Finally, to find the area of the square, we square the side length.
Area = (Side)^2 ≈ (16.97 cm)^2 ≈ 288 cm^2

Therefore, the area of the square that can be inscribed in a circle of radius 12 cm is approximately 288 square cm, which corresponds to option A.