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.
image of a circle divided into equal slices and rearranged to resemble a parallelogram
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 = 3 ) cm.
answer the following.
when applicable, use the symbol ( pi ) for your answers. (do not give decimal answers.)
part 1:
the length of the base of the \parallelogram\ is choose one the circumference of the circle.
the circumference of the circle is cm.
so, the length of the base of the \parallelogram\ is ( b = ) cm.
part 2:
the height of the \parallelogram\ is choose one the radius of the circle.
so, the height of the \parallelogram\ is ( h = ) cm.
part 3:
use the parts above to find the area of the \parallelogram\.
area of \parallelogram\ = ( \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} = pi r )
• ( \text{area} = 3pi r^2 )
• ( \text{area} = 6pi r^2 )
• ( \text{area} = 9pi r )
• ( \text{area} = pi r^2 )

Explanation:

Step1: Calculate circle circumference

Circumference formula: $C = 2\pi r$. For $r=3$, $C=2\pi(3)=6\pi$.

Step2: Find base of parallelogram

Base is half the circumference: $b=\frac{6\pi}{2}=3\pi$.

Step3: Determine height of parallelogram

Height equals radius: $h=r=3$.

Step4: Calculate parallelogram area

Area formula: $Area = b \times h = 3\pi \times 3 = 9\pi$.

Step5: Identify circle area equation

Area of circle equals parallelogram area, so $Area=\pi r^2$.

Answer:

Part1: half; $6\pi$; $3\pi$
Part2: equal to; $3$
Part3: $9\pi$
Part4: Area = $\pi r^2$