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A torus with major radius $R$ and minor radius $r$ has volume $2 π^{2} R r^{2}$

Example

Let $R > r > 0$ . Rotating the disc $(x - R)^{2} + y^{2} \leq r^{2}$ about the $y$ -axis produces a ring torus of volume $2 π^{2} R r^{2}$ .

Facts & Assumptions

Given: Reals $R > r > 0$ and the stated generating disc.

[F1]

A washer solid with outer radius $f$ and inner radius $g$ has volume $π \int (f^{2} - g^{2})$ (The washer formula for a solid of revolution between two nonnegative profiles).

[F2]

A closed disc of radius $r \geq 0$ has Jordan content $π r^{2}$ (A closed disc of radius $r \geq 0$ has Jordan content $π r^{2}$ ).

Verification

technique · direct

1.1

At height $y \in [- r, r]$ , put $q (y) = \sqrt{r^{2} - y^{2}}$ . The outer and inner radii are $R + q (y)$ and $R - q (y)$ ; both are nonnegative because $R > r \geq q (y)$ .

givenalgebra

2.1

By [F1], the washer area is $π ((R + q)^{2} - (R - q)^{2}) = 4 π R q$ , so the torus volume is $4 π R \int_{- r}^{r} \sqrt{r^{2} - y^{2}} d y$ .

step 1.1F1algebra

3.1

The integral in step 2.1 is the area under the upper semicircle of radius $r$ , hence half the disc content [F2], namely $π r^{2} / 2$ . Thus the volume is $2 π^{2} R r^{2}$ .

step 2.1F2algebra∎

Depends on

Used by

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Sources

Sigurd Angenent, Math 221 lecture notes, Chapter 8 §4 (standard reference, not scraped)

A torus with major radius RR and minor radius rr has volume 2π2Rr22π2Rr2