CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-24

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A right circular cylinder of radius $R$ and height $h$ has volume $π R^{2} h$

Statement

A right circular cylinder of radius $R \geq 0$ and height $h \geq 0$ has volume $π R^{2} h$ .

Facts & Assumptions

Given: Nonnegative reals $R, h$ , with the cylinder presented as the solid of revolution of the constant profile $R$ on $[0, h]$ .

[F1]

A solid of revolution with profile $f$ has volume $π \int_{a}^{b} f (x)^{2} d x$ (The disc formula for the volume of a solid of revolution).

Proof

technique · direct

1.1

Apply [F1] to the constant profile $f (x) = R$ on $[0, h]$ .

givenF1

2.1

The integral of the constant $π R^{2}$ over an interval of length $h$ is $π R^{2} h$ . This remains zero when $R = 0$ or $h = 0$ .

step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

23 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Sigurd Angenent, Math 221 lecture notes, Chapter 8 §3 (standard reference, not scraped)

A right circular cylinder of radius RR and height hh has volume πR2hπR2h