Radian measure — definition, conversion formula degrees ↔ radians: Worked examples

This page is the drill hall for radian measure. The parent note built the definition $θ = s / r$ and the master key $18 0^{\circ} = π$ rad. Here we throw every kind of question at you and work each one to the bone. Nothing new is assumed — if a symbol appears, it was earned in the parent note or is rebuilt below.

The scenario matrix

Every question this topic can throw belongs to one of these cells. Each row is covered by at least one worked example below.

Cell	Case class	What makes it tricky	Example
A	Degrees → radians, "nice" angle	pick the shrinking factor $\frac{π}{180}$	Ex 1
B	Radians → degrees, $π$ -form	the $π$ must cancel	Ex 2
C	Radians → degrees, decimal (no $π$ )	use $1 rad \approx 57. 3^{\circ}$	Ex 3
D	Negative angle	sign rides along untouched	Ex 4
E	Bigger than a full turn (> $36 0^{\circ}$ or > $2 π$ )	co-terminal reduction	Ex 5
F	Zero / degenerate input	arc collapses, formula still holds	Ex 6
G	Real-world word problem (arc length)	must convert before $s = r θ$	Ex 7
H	Real-world word problem (angular speed)	rad/s vs deg/s, Circular Motion	Ex 8
I	Exam twist: solve backwards for $r$ or $θ$	rearrange the definition	Ex 9
J	Limiting behaviour ( $θ \to 0$ )	link to Small Angle Approximation	Ex 10

Cell A — Degrees → radians (nice angle)

Cell B — Radians → degrees ( $π$ -form)

Cell C — Radians → degrees (decimal, no $π$ )

Cell D — Negative angle

Cell E — Bigger than a full turn (co-terminal)

Cell F — Zero / degenerate input

Cell G — Word problem, arc length

Cell H — Word problem, angular speed

Cell I — Exam twist (solve backwards)

Cell J — Limiting behaviour ( $θ \to 0$ )

Worked example Ex 10. For a tiny angle, compare the arc

$s$ and the straight chord across it on a radius- $1$ circle. Use $θ = 0.02$ rad. Forecast: For a very small angle the curved arc and the straight chord are almost indistinguishable, so $s \approx sin θ$ — the whole point of the Small Angle Approximation. Expect the two numbers to agree to several decimals.

Step 1 — arc length on the unit circle. Why this step? With $r = 1$ , $s = r θ = θ$ exactly — radians make the arc equal the angle. This is only true in radians. $s = 1 \times 0.02 = 0.02$

Step 2 — compare with $sin θ$ (the vertical rise, a stand-in for the chord's half-height). Why this step? As $θ \to 0$ , $sin θ \to θ$ ; this equality is what makes $\tfrac{d}{d\theta}\sin\theta = \cos\theta$ clean. $sin (0.02) \approx 0.0199987$

Verify: $s = 0.02$ vs $sin θ \approx 0.0199987$ — they agree to $4$ decimal places. As $θ \to 0$ the ratio $\frac{s i n θ}{θ} \to 1$ , so arc $\approx$ chord. This limit requires radians. ✓

Recall Which factor, which direction? (self-test)

Convert $15 0^{\circ}$ to radians ::: $150 \times \frac{π}{180} = \frac{5 π}{6}$ rad. Convert $\frac{11 π}{6}$ rad to degrees ::: $\frac{11 π}{6} \times \frac{180}{π} = 33 0^{\circ}$ . $78 0^{\circ}$ points the same way as which angle in $[0, 36 0^{\circ})$ ? ::: $6 0^{\circ}$ (subtract $36 0^{\circ}$ twice). Arc of $1.75$ rad, length $14$ cm — radius? ::: $r = 14/1.75 = 8$ cm. Why must you convert before using $s = r θ$ ? ::: Because $s = r θ$ is the rearranged radian definition; it only holds in radians.

Connections

Advanced Trigonometry — parent chapter
Arc Length and Sector Area — Ex 6, 7, 9
Unit Circle — Ex 5, 10
Trigonometric Functions of Any Angle — Ex 4 (negative), Ex 5 (co-terminal)
Small Angle Approximation — Ex 10
Derivatives of Trig Functions — why Ex 10's limit matters
Circular Motion — Ex 8 (rad/s)

Worked examples — Radian measure — definition, conversion formula degrees ↔ radians