Question

suppose the bottom surface, or base, of a cylinder is divided into 16 congruent sectors. then the sectors are rearranged as shown. open this geogebra activity to divide the circle into even more sectors and rearrange the sectors for yourself.
complete the statements to compare the areas of the two figures and find the area of the base of the cylinder.
when the sectors of the circle are rearranged, they are close to forming a ▼ with a length of ▼ units and a width of r units.
the area of the rearranged figure is ▼ to that of the circle. so, the area of the circular base is ▼ square units.
select the correct answer from each drop - down menu.

Explanation:

Step1: Recall the formula for the area of a circle

The area of a circle is \(A = \pi r^{2}\).

Step2: Analyze the rearrangement of the circle's sectors

When the sectors of a circle are rearranged, the resulting figure (approximating a parallelogram - like shape) has a length (base of the parallelogram - like shape) equal to half of the circumference of the circle (\(C=\pi r\)) and a width equal to the radius \(r\) of the circle. The area of a parallelogram is \(A = \text{base}\times\text{height}\). Here, base\(=\pi r\) and height\( = r\), so the area of the rearranged figure is \(\pi r\times r=\pi r^{2}\).

Answer:

The area of the circular base is \(\pi r^{2}\) square units. The area of the rearranged figure is equal to that of the circle. When the sectors of the circle are rearranged, they are close to forming a parallelogram with a length of \(\pi r\) units and a width of \(r\) units.