Question

course activity: explaining volume formu... 7 of 12 save 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. image of a circle divided into 16 sectors and the rearranged sectors forming a shape complete the statements to compare the areas of the two figures and find the area of the base of the cylinder. select the correct answer from each drop - down menu. when the sectors of the circle are rearranged, they are close to forming a drop - down with a length of drop - down units and a width of r units. the area of the rearranged figure is drop - down to that of the circle. so, the area of the circular base is drop - down square units.

Explanation:

Step1: Analyze the rearranged figure

When a circle is divided into many congruent sectors and rearranged, the shape approaches a rectangle. The length of this rectangle is half of the circumference of the circle. The formula for the circumference of a circle is \( C = 2\pi r \), so half of it is \( \pi r \).

Step2: Compare the areas

The area of the rearranged figure (rectangle) is equal to the area of the original circle because we are just rearranging the sectors, not changing the total area.

Step3: Calculate the area of the base (circle)

The area of a rectangle is length times width. Here, length is \( \pi r \) and width is \( r \), so the area is \( \pi r\times r=\pi r^{2} \).

Answer:

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