GMY-30

Sphere solid angle

Projected steradians.

Reading speed

GoverningSphere solid angle

Governing equation

\\Omega=A/R^2

$A \in \left[0.1, 80\right]$

where

A: A (m²)
R: R (m)
Om: Sphere solid angle (sr)

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

\\Omega=A/R^2

$A \in \left[0.1, 80\right]$

$A = 0.1$ $Om = 0.011$

Along the curve

②

Domain of the variable — blue

A_{\min} = 0.1,\quad A_{\max} = 80

$A \in \left[0.1, 80\right]$

$A = 0.1$ $A = 0.1$

Along the curve

③

Response curve — green

\\Omega=A/R^2

$A \in \left[0.1, 80\right]$

$A = 0.1$ $Om = 0.011$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}Om}{\mathrm{d}A}

$A \in \left[0.1, 80\right]$

$A = 0.603$ $Om^{\prime} = 0.111$

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-30 — Sphere solid angle) is the form associated with Sphere solid angle. Working symbols: $A$ , $R$ $\rightarrow$ $Om$ . Projected steradians. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

Om

from

A

R

in Geometry via

\\Omega=A/R^2

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

Live realistic example

In symbols

Live case. Given

A = 12.000\,\mathrm{m^{2}}

R = 3.000\,\mathrm{m}

, the governing relation

\\Omega=A/R^2

yields

Om = 1.333\,\mathrm{sr}

. 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

A A12 m²R R3 m

Outputs

Sphere solid angle Om1.333 sr

Reading speed

Watch on YouTube

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Narration of this film

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

Projected steradians. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube