Question

the circle below is divided into equal slices.
the figure to the right of the circle shows the slices rearranged.
use the slider and observe what happens to the figure on the right as the number of slices increases.

as the number of slices increases, the rearranged figure more closely resembles a parallelogram.
for this problem, we will assume that the final rearranged figure is a parallelogram.
(we will refer to this parallelogram with quotes in the statements below.)
suppose the circle has radius ( r = 4 ) cm.
answer the following.
when applicable, use the symbol ( pi ) for your answers. (do not give decimal answers.)

part 1:
the height of the \parallelogram\ is choose one the radius of the circle.
so, the height of the \parallelogram\ is ( h = square ) cm.

part 2:
the length of the base of the \parallelogram\ is choose one the circumference of the circle.
the circumference of the circle is ( square ) cm.
so, the length of the base of the \parallelogram\ is ( b = square ) cm.

part 3:
use the parts above to find the area of the \parallelogram\.

( \text{area of \parallelogram\} = square ) ( \text{cm}^2 )

part 4:
the circle was rearranged to make the \parallelogram\.
so the area of the circle is equal to the area of the \parallelogram\.
using this fact, choose the equation that gives the area of the circle in terms of ( r ).
( \bigcirc \text{area} = pi r^2 )
( \bigcirc \text{area} = 2pi r )
( \bigcirc \text{area} = pi r )
( \bigcirc \text{area} = 8pi r^2 )
( \bigcirc \text{area} = 4pi r^2 )

Explanation:

Step1: Determine height of "parallelogram"

The height equals the circle's radius. Given \(r = 4\) cm, so \(h = 4\) cm.

Step2: Calculate circle's circumference

Circumference formula: \(C = 2\pi r\). Substitute \(r = 4\): \(C = 2\pi(4) = 8\pi\) cm. The base of the "parallelogram" is half the circumference: \(b = \frac{8\pi}{2} = 4\pi\) cm.

Step3: Compute area of "parallelogram"

Area formula for parallelogram: \(Area = base \times height\). Substitute \(b = 4\pi\) and \(h = 4\): \(Area = 4\pi \times 4 = 16\pi\) cm².

Step4: Identify circle area formula

Since circle area equals "parallelogram" area, the formula is \(Area = \pi r^2\).

Answer:

Part1: equal to; 4
Part2: half; \(8\pi\); \(4\pi\)
Part3: \(16\pi\)
Part4: Area = \(\pi r^2\)