Final answer:
To compare the areas of two circles associated with a square, one uses the circle area formula πr². The inscribed circle's radius is half the square's side, and the circumscribed circle's radius is half the square's diagonal length. The ratio of the areas is 2:1.
Step-by-step explanation:
The student is asking to compare the areas of two circles related to a square: one circumscribed about the square and the other inscribed within the same square. To find the ratio of their areas, we need to use the formula for the area of a circle, πr², where π is approximately 3.14159 and r is the radius of the circle.
The side of the square, a, is 6 units. For the inscribed circle (circle 2), the diameter is equal to the side of the square, so the radius (r) is 3 units. Thus, its area is π(3)² = 9π square units. For the circumscribed circle (circle 1), the diameter is equal to the diagonal of the square. Since the square has sides of 6 units, by the Pythagorean theorem, the diagonal is √(6² + 6²) = √(72) = 6√2. The radius of circle 1 is therefore 3√2 units, giving an area of π(3√2)² = 18π square units. The ratio of the area of circle 1 to circle 2 is 18π / 9π = 2:1.