Final answer:
The circumference and area of a circle can be approximated within a square where the diameter matches the side length of the square. The circumference is given as 2πr, while the area of the circle is πr². The formulas use π (approximately 3.14159) and must match the significant figures of the given radius.
Step-by-step explanation:
The question involves justifying the formulas for the circumference and area of a circle. When fitting a circle inside a square such that the diameter equals the side of the square (a = 2r), the circumference of the circle is less than the perimeter of the square but more than twice the square's side. Consequently, since the side length a is twice the radius r, the circumference, which is the distance around the circle, can be approximated as 2πr.
Regarding the area, the circle's area inside the square will be less than the square's area (a²) but more than half of it. If we consider the square's area as being 4r², then three-quarters of this would approximate to 3r² which is close to πr², the actual area of a circle. Therefore, the formula for calculating the area is correctly expressed as A = πr².
Remember, when working with equations involving physical dimensions, the units of measurement can guide you to the correct formula. For example, a circle's area will have units of square meters (m²), indicating that the formula will involve the radius squared. This harmonizes with the relationship A = πr², where π is approximately 3.14159.
Furthermore, when calculating quantities like the area, the number of significant figures needed in the result should match the number of significant figures in the given radius. If the radius is given as 2.0 meters, with two significant figures, the calculated area should also be presented with two significant figures, i.e. A = 4.5 m².