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G-GMD.1 Core M2-037-R09-V02

Explain circumference, area, and volume formulas using informal arguments such as dissection, Cavalieri, and limits.

Build volume by accumulating cross-sections

Problem

A cone is approximated by \(n\) layers whose \(k\)th sample radius is \(R(k/n)\), so its slice area is \(\pi~R^{2}(k/n)^{2}\) and thickness is \(h/n\). Form the volume sum and use the limiting sum-of-squares factor to derive the cone volume formula.

Cone divided into n layers, with kth sample radius R(k/n), slice area πR²(k/n)², and thickness h/n. Open full size
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Four variants of this problem type
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
Build volume by accumulating cross-sections