GMY-23

Torus volume

Ring torus.

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GoverningTorus volume

Governing equation

V=2\\pi^2 R r^2

$R \in \left[0.4, 8\right]$

where

R: R (m)
r: r (m)
V: Torus volume (m³)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

V=2\\pi^2 R r^2

$R \in \left[0.4, 8\right]$

$R = 0.4$ $V = 1.263$

Along the curve

②

Domain of the variable — blue

R_{\min} = 0.4,\quad R_{\max} = 8

$R \in \left[0.4, 8\right]$

$R = 0.4$ $R = 0.4$

Along the curve

③

Response curve — green

V=2\\pi^2 R r^2

$R \in \left[0.4, 8\right]$

$R = 0.4$ $V = 1.263$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}V}{\mathrm{d}R}

$R \in \left[0.4, 8\right]$

$R = 0.448$ $V^{\prime} = 3.158$

Along the curve

Lecture brief

Historical brief

Pythagoras, Heron, spherical trigonometry and the sphere’s volume are the measuring arts that predate calculus. The sheets compute length, area and angle on plane and sphere. This sheet (GMY-23 — Torus volume) is the form associated with Torus volume. Working symbols: $R$ , $r$ $\rightarrow$ $V$ . Ring torus. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

V

from

R

r

in Geometry via

V=2\\pi^2 R r^2

Ring torus. Use it when a real geometry question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

R = 2.000\,\mathrm{m}

r = 0.400\,\mathrm{m}

, the governing relation

V=2\\pi^2 R r^2

yields

V = 6.317\,\mathrm{m^{3}}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

R R2 mr r0.4 m

Outputs

Torus volume V6.317 m³

Reading speed

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Ring torus. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube