Final answer:
The student's question relates to geometric properties of circles within squares whereby the diameter of the circle is equal to the side of the square. The concepts discussed involve relationships between perimeter and area of both shapes, and the approximation of arc lengths in circular geometry.
Step-by-step explanation:
The question involves mathematical concepts surrounding circles and geometry. To fit a circle inside a square such that the diameter is equal to the side length of the square, we express this as a = 2r where a is the side length of the square and r is the radius of the circle. This implies that the perimeter of the circle, which is given by the formula 2πr, should be significantly less than the square's perimeter of 4a but larger than 2a.
The area considerations follow a similar logic. The area of the circle is πr² which will be less than the area of the square, a² or 4r². If we estimate that the area of the circle is approximately three-quarters of the area of the square, we are led to the approximation that 3r² ~ πr².
A key concept in this question is the approximation of a small arc length as being equivalent to its baseline in a circle (AB ~ arc AB), which is used in various problems in geometry dealing with arcs and circles.