Explain circumference, area, and volume formulas using informal arguments such as dissection, Cavalieri, and limits.
Recover circle area by rearranging sectors
Problem
A circle of radius \(4\) is cut into many equal sectors and arranged alternately. Describe the limiting parallelogram-like shape, derive its base from \(\text{half}~\text{the}~\text{circumference}\) and its height from the radius, then calculate the circle's area.
The rearrangement is useful because it trades a curved region for a near-parallelogram without changing area. We’ll track which arcs form one long base, relate that base to half the original circumference, identify the radius as the perpendicular height, and connect base times height back to the circle.
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