Problem preview

G-GMD.1 Warmup M2-037-A11-V03

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

Choose the informal argument that supports a geometric formula

Problem

Explain the formula \(V~=~(1/3)\pi~r^{2}h\) for a cone by comparing it with a cylinder having the same base radius and height. State the cylinder volume and the cone-to-cylinder volume relationship.

Answer choices

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Compare the cone to a cylinder with the same base radius and slant height, since matching the side lengths explains the one-third factor.

Stack congruent circular layers from the base to the top, so every slice has area πr² and the volume is one-third πr²h.

Dissect the cone’s base and rearrange it into a rectangle, which shows the cone’s volume must be one-third πr²h.

Compare the cone to a cylinder with the same base radius and height, then use that the cone has one-third of the cylinder’s volume.

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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: Choose the informal argument that supports a geometric formula