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 to 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 = 9 ) 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 ).

• ( \text{area} = 18pi r^2 )
• ( \text{area} = 9pi r^2 )
• ( \text{area} = pi r^2 )
• ( \text{area} = pi r )
• ( \text{area} = 81pi r )

Explanation:

Step1: Determine height of parallelogram

The height equals the circle's radius $r=9$ cm.

Step2: Calculate circumference of circle

Circumference formula: $C=2\pi r$. Substitute $r=9$: $C=18\pi$ cm.

Step3: Find base of parallelogram

Base is half the circumference: $b=\frac{18\pi}{2}=9\pi$ cm.

Step4: Compute area of parallelogram

Area formula: $Area = b \times h$. Substitute $b=9\pi$, $h=9$: $Area=81\pi$ cm².

Step5: Identify circle area formula

Circle area equals parallelogram area, so formula is $Area=\pi r^2$.

Answer:

Part1: equal to; 9
Part2: half; 18π; 9π
Part3: 81π
Part4: Area = πr²