GMY-07

Sphere volume

V = 4π r³ / 3. The solid of constant curvature.

Reading speed

MetricEuclid XII

Governing equation

V=\dfrac{4}{3}\pi r^3

$r \in \left[0.1, 20\right]$

where

r: Radius (m)
V: 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=\dfrac{4}{3}\pi r^3

$r \in \left[0.1, 20\right]$

$r = 0.1$ $V = 0.004$

Along the curve

②

Domain of the variable — blue

r_{\min} = 0.1,\quad r_{\max} = 20

$r \in \left[0.1, 20\right]$

$r = 0.1$ $r = 0.1$

Along the curve

③

Response curve — green

V=\dfrac{4}{3}\pi r^3

$r \in \left[0.1, 20\right]$

$r = 0.1$ $V = 0.004$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}V}{\mathrm{d}r}

$r \in \left[0.1, 20\right]$

$r = 0.225$ $V^{\prime} = 0.703$

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-07 — Sphere volume) is the form associated with Euclid XII. Working symbols: $r$ $\rightarrow$ $V$ . Cavalieri / Archimedes: the sphere is 2/3 of its circumscribed cylinder.

Purpose

Purpose: compute

V

from

r

in Geometry via

V=\dfrac{4}{3}\pi r^3

V = 4π r³ / 3. The solid of constant curvature. 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}

, the governing relation

V=\dfrac{4}{3}\pi r^3

yields

V = 33.5103\,\mathrm{m^{3}}

. A sphere growing with r. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Radius r2 m

Outputs

Volume V33.5103 m³

Reading speed

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Narration of this film

A sphere growing with r.

Cavalieri / Archimedes: the sphere is 2/3 of its circumscribed cylinder.

Reading speed

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