Inscribed Square Area Calculation

The problem asks for the area of the largest possible square drawn inside a circle with a radius of 12 cm.

Relating Circle and Square Dimensions

For a square inscribed in a circle, the diagonal of the square is equal to the diameter of the circle.

• Circle radius ($r$) = 12 cm.
• Circle diameter ($d$) = $2 \times r = 2 \times 12 \text{ cm} = 24 \text{ cm}$.
• Therefore, the diagonal of the inscribed square ($D$) = 24 cm.

Calculating Square Area

Let the side length of the square be $s$. The area of the square is given by $A = s^2$.

Using the Pythagorean theorem on the square's diagonal ($D$) and sides ($s$): $s^2 + s^2 = D^2$ $2s^2 = D^2$

Substitute the value of the diagonal:

$2s^2 = (24 \text{ cm})^2$

$2s^2 = 576 \text{ cm}^2$

To find the area ($s^2$), divide by 2:

$s^2 = \frac{576 \text{ cm}^2}{2}$

$s^2 = 288 \text{ cm}^2$

The area of the square is $s^2$, which is 288 sq. cm.