CLASSICAL-MECHANICS

Radians, Steradians, and the Small Angle Approximation

IKey Formulas

Formula	Name	Description
$\theta = \dfrac{s}{r}$	Radian definition	Arc length over radius (pure ratio)
$360^\circ = 2\pi \text{ rad}$	Conversion	One full turn; $1\text{ rad} \approx 57.3^\circ$
$\sin\theta \approx \theta \approx \tan\theta$	Small angle approximation	Valid for small $\theta$ in radians
$\Omega = \dfrac{A}{r^2}$	Steradian definition	Solid angle: area over radius squared (sphere $= 4\pi$ sr)

IIInteractive Visualizations

Visualization 1 — What Is a Radian?

One radian is the angle whose arc equals the radius, making $\theta = s/r$ a pure number.

Angle θ (rad) 1.00

💡 The ratio $s/r$ is identical on every circle, so a full turn is $2\pi$ rad no matter the radius.

Visualization 2 — When sin θ ≈ θ ≈ tan θ

Near zero the curves share the $45^\circ$ line $y=\theta$ ; the probe reads how fast they fan apart.

Probe θ (rad) 1.00

💡 The zone edge, $0.2$ rad, is about $11.5^\circ$ ; past it the straight-line approximation visibly fails.

Visualization 3 — Tiny Angles, Huge Distances

With the baseline fixed, distance is locked to the angle by $d = \text{baseline}/\theta$ — smaller angle, farther star.

Angle (arcseconds) 1.0

💡 A $1$ AU baseline subtending $1''$ defines the parsec: $1\text{ pc} \approx 2.06\times10^{5}$ AU $\approx 3.26$ light-years.

IIIQuiz Questions

Question 1

An arc of length $s = 3$ m is drawn on a circle of radius $r = 1.5$ m. What is the angle subtended, in radians?

✅ Correct! θ = s/r = 3/1.5 = 2 rad.

❌ Not quite. Use θ = s/r — divide the arc length by the radius.

Show solution

Solution:

The radian is defined as arc length over radius: $\theta = \frac{s}{r} = \frac{3}{1.5} = 2 \text{ rad}.$

The result is a pure ratio (length over length), so it is dimensionless.

Question 2

A student writes $\sin\theta \approx \theta$ and plugs in $\theta = 0.1$ degrees, expecting to get approximately $0.1$ . Is this use of the small angle approximation valid?

✅ Correct! The approximation requires θ in radians; degrees carry an arbitrary scaling factor.

❌ Not quite. sin(0.1°) ≈ 0.0017, not 0.1 — the rule fails in degrees.

Show solution

Solution:

The approximation $\sin\theta \approx \theta$ holds only in radians, because radians are dimensionless ratios with no arbitrary scaling factor.

In degrees, $\sin(0.1^\circ) \approx 0.0017$ , which is nowhere near $0.1$ . To use the approximation you must first convert: $0.1^\circ = 1.75\times10^{-3}$ rad, and indeed $\sin(1.75\times10^{-3}) \approx 1.75\times10^{-3}$ .

Question 3

A baseline of $1$ AU subtends an angle of $2$ arcseconds at a distant star. Using $1'' = 4.85\times10^{-6}$ rad and the small angle relation $d = \text{baseline}/\theta$ , approximately how far away is the star, in AU?

✅ Correct! d = 1 AU / (9.7×10⁻⁶) ≈ 1.0×10⁵ AU.

❌ Not quite. Convert 2″ to radians first, then divide the baseline by θ.

Show solution

Solution:

Convert the angle: $\theta = 2'' = 2 \times 4.85\times10^{-6} = 9.7\times10^{-6}$ rad.

Apply $d = \text{baseline}/\theta$ : $d = \frac{1 \text{ AU}}{9.7\times10^{-6}} \approx 1.0\times10^{5} \text{ AU}.$

A $1''$ angle gives $2.06\times10^{5}$ AU (one parsec); doubling the angle halves the distance.

Question 4

The radian is the 2D angle defined as arc/radius, and the full circle measures $2\pi$ rad. Its 3D cousin, the steradian, is the solid angle defined as $\Omega = A/r^2$ . How many steradians does a complete sphere subtend?

✅ Correct! Ω = 4πr²/r² = 4π sr.

❌ Not quite. Divide the sphere's area 4πr² by r² so the radius cancels.

Show solution

Solution:

The solid angle is $\Omega = A/r^2$ . The total surface area of a sphere is $A = 4\pi r^2$ , so $\Omega_{\text{sphere}} = \frac{4\pi r^2}{r^2} = 4\pi \text{ sr}.$

The radius cancels, just as it does for the radian. Note the parallel: full circle $= 2\pi$ rad, full sphere $= 4\pi$ sr. The answer $4\pi r^2$ is wrong because the $r^2$ must cancel — a solid angle is dimensionless.

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