Question

explore the relationship between the radius and area of a circle by following these steps. 1. press continue to rearrange the sectors of the circle to form a shape resembling a parallelogram.

Explanation:

Step1: Recall circle area derivation

When we rearrange the sectors of a circle (divided into equal - angled sectors, like the 8 - sector circle here) to form a shape similar to a parallelogram:

• The circumference of the circle is $C = 2\pi r$. When we form the parallelogram - like shape, the length of the "base" of this shape is approximately half of the circumference of the circle. So the length of the base $b=\pi r$.
• The height of this parallelogram - like shape is approximately equal to the radius of the circle, $h = r$.

Step2: Use the formula for the area of a parallelogram

The formula for the area of a parallelogram is $A=b\times h$.
Substitute $b = \pi r$ and $h = r$ into the formula:
$A=\pi r\times r=\pi r^{2}$
This shows that the area of a circle $A=\pi r^{2}$, and we can see the relationship between the radius $r$ and the area $A$: the area is proportional to the square of the radius, with the constant of proportionality being $\pi$.

Answer:

The area of a circle $A$ is related to its radius $r$ by the formula $A = \pi r^{2}$, meaning the area is proportional to the square of the radius. When we rearrange the circle's sectors to form a parallelogram - like shape, the base of the parallelogram is $\pi r$ (half of the circle's circumference) and the height is $r$ (the circle's radius), and using the parallelogram area formula $A = base\times height$ gives us the circle's area formula.