GMY-03

Sphere volume

Also 4πR².

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Geometrysphere

Governing equation

V=4\\pi R^3/3

$R \in \left[0.05, 10\right]$

where

R: R (m)
V: V (m³)
A: A (m²)

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=4\\pi R^3/3

$R \in \left[0.05, 10\right]$

$R = 0.05$ $V = 5.24\times 10^{-4}$

Along the curve

②

Domain of the variable — blue

R_{\min} = 0.05,\quad R_{\max} = 10

$R \in \left[0.05, 10\right]$

$R = 0.05$ $R = 0.05$

Along the curve

③

Response curve — green

V=4\\pi R^3/3

$R \in \left[0.05, 10\right]$

$R = 0.05$ $V = 5.24\times 10^{-4}$

Along the curve

④

Rate of change — amber

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

$R \in \left[0.05, 10\right]$

$R = 0.113$ $V^{\prime} = 0.176$

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-03 — Sphere volume) is the form associated with sphere. Working symbols: $R$ $\rightarrow$ $V$ , $A$ . Also 4πR².

Purpose

Purpose: compute

V

A

from

R

in Geometry via

V=4\\pi R^3/3

Also 4πR². 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 = 1.000\,\mathrm{m}

, the governing relation

V=4\\pi R^3/3

yields

V = 4.1888\,\mathrm{m^{3}}

A = 12.5664\,\mathrm{m^{2}}

. Also 4πR². Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

R R1 m

Outputs

V V4.1888 m³
A A12.5664 m²

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Narration of this film

Also 4πR².

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