Question

question 3 of 40
below is the justification for the formula for area of a circle. which word, when placed in the blank, best completes this argument?

• the circumference of a circle is given by the formula c = 2πr, where r is the radius.
• if the circle is divided into equally sized sectors, the sectors can be rearranged to form a shape that approximates a parallelogram.
• the base of the parallelogram is ____, and the height is r.
• because the area of a parallelogram is equal to the product of the base and the height, the area is πr².
• therefore, the area of a circle is given by the formula πr².

a. the radius, r
b. half the radius, or 1/2 r
c. half the circumference, or πr
d. the circumference, or 2πr²

Explanation:

Step1: Recall circle - parallelogram transformation

When a circle is divided into sectors and rearranged into an approximate parallelogram, the arc - lengths of the sectors contribute to the base of the parallelogram.

Step2: Relate circle's circumference to parallelogram's base

The full - circumference of the circle is \(C = 2\pi r\). Since the sectors are rearranged to form the base of the parallelogram, and half of the sectors' arc - lengths make up one side of the base, the base of the parallelogram is half of the circumference of the circle. So the base of the parallelogram is \(\frac{2\pi r}{2}=\pi r\).

Answer:

C. half the circumference, or \(\pi r\)