Back to Explain circle area and volume formulas — Problem 2 · Task Set 33

Exercises: Informal Arguments for Volume and Area Formulas

Grade 10·24 problems·~30 min·Common Core Math - HS Geometry·standard·hsg-gmd-a-1

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Fluency Practice

In the wedge-dissection argument for circle area, a circle of radius $r$ is cut into many thin sectors and rearranged into a shape resembling a rectangle. Which dimensions does this near-rectangle have, and what area does it give?

Width $= 2\pi r$ , height $= r$ ; area $= 2\pi r^2$ — so the circle's area is $2\pi r^2$

Width $= \pi r$ , height $= r$ ; area $= \pi r^2$ — confirming $A = \pi r^2$

Width $= 2r$ , height $= \pi$ ; area $= 2\pi r$ — so the circle's area equals its circumference

Width $= r$ , height $= r$ ; area $= r^2$ — since $\pi$ cancels out in the rearrangement