Problem preview

G-GMD.1 Core M2-037-R02-V04

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 \(10\) 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.

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Use base 10π and diameter 20 as the height, giving 200π square units.

Use full circumference 20π as the base and height 10, giving 200π square units.

Use base 10π and height 10, then double 100π to get 200π square units.

The rearrangement approaches a parallelogram with base πr = 10π and height 10, so its area is 100π square units.

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Curriculum context

Course: Math II
Standard: G-GMD.1
Category: Geometry
Domain: Geometric Measurement and Dimension
Objective: Explain circumference, area, and volume formulas using informal arguments such as dissection, Cavalieri, and limits.
Problem type: Recover circle area by rearranging sectors