Final answer:
The area of the square circumscribed about a circle with a 6-foot radius is 144 square feet, calculated by squaring the side length of the square (12 feet), which is twice the radius of the circle.
Step-by-step explanation:
The question asks for the area of a square circumscribed about a circle with a 6 foot radius. To find the area of the square, we need to understand that the diameter of the circle is equal to the length of a side of the square. Since the radius of the circle is 6 feet, the diameter is 12 feet (2 times the radius). Therefore, each side of the square is also 12 feet.
The area of a square is calculated by squaring the side length (a), so the area in this case is:
Area = a²
= (12 feet)²
= 144 square feet.
This formula is supported by the idea that the area of a square is always a squared value (length × width) and in a square, these dimensions are equal.
As a helpful reference, consider that a circle inscribed within the square would naturally have a smaller area than the square, and since we know the side of the square, we can easily compute the square's area.
In conclusion, the area of the square is 144 square feet, which is obtained by squaring the length of the side (12 feet).