Final answer:
The correct answer is b: as the number of sections increases, the length of the segments decreases while the width increases.
Step-by-step explanation:
When a circle with radius r is divided into more and more congruent sections, the length of the segments (arc length) decreases while the width of the segments increases. This occurs because as the number of sections increases, each section spans a smaller central angle, leading to a shorter arc length for each segment. However, because the circumference of the circle remains constant (the circumference of a circle is given by 2πr, where π is approximately 3.14159), the width of the segments at the periphery of each section increases to maintain the circumference. Therefore, the correct statement is (b) the length of the segments decreases while the width increases.
About those circles and spheres: to fit a circle inside of a square such that its diameter is equal to the side length of the square (a = 2r), we consider the circle's perimeter which should be somewhat close to 6r, as it will be less than the perimeter of the square (4a) but greater than a diameter line across the square (2a).
In addition, the area of the circle within the square is smaller than that of the square but larger than half of it. The full area of the square is 4r² and roughly three-quarters of this area approximates the area of the circle, which is πr².