Angles in rotation are measured in radians, and angular velocity (ω) tells how fast the angle changes; a point's speed is v = rω.

\displaystyle \Delta\theta = \frac{\Delta s}{r}

\displaystyle \omega = \frac{\Delta\theta}{\Delta t}

\displaystyle v = r\omega

Rotation angle (Δθ) — how far something has turned. It's arc length divided by radius.
Arc length (Δs) — distance traveled along the circle.
Radius of curvature (r) — radius of the circular path.
Radian (rad) — the unit for rotation angles. One revolution = 2π rad. Radians have no real units (meters ÷ meters), so rad/s works out as 1/s.
Angular velocity (ω) — rate of rotation, in rad/s. Its direction is just clockwise or counterclockwise.
Tangential (linear) speed (v) — speed of a point moving around the circle, pointing along the tangent.

Conversions

From	To	Multiply by
Revolutions	Radians	2π
Degrees	Radians	π / 180
Radians	Degrees	180 / π (1 rad ≈ 57.3°)
rev/min (rpm)	rad/s	2π / 60

Common angles: 30° = π/6, 60° = π/3, 90° = π/2, 120° = 2π/3, 135° = 3π/4, 180° = π, 360° = 2π.

Key ideas

Every point on a spinning object turns through the same angle in the same time, so it has the same ω.
Points farther from the center travel farther, so they move faster: v = rω. The tip of a bat moves faster than the handle.
For the same road speed, a bigger wheel spins slower (ω = v/r).
After one full revolution (2π rad), every point is back where it started.

Worked examples

Problem	Setup	Answer
Car tire, r = 0.300 m, car at 15.0 m/s (Ex. 6.1)	ω = v/r = 15.0 / 0.300	50.0 rad/s
Earth-mover tire, r = 1.20 m, same speed	ω = 15.0 / 1.20	12.5 rad/s

Common mistakes: using degrees in v = rω or Δs = rΔθ (must be radians); forgetting the 2π when converting revolutions; thinking the outer edge has a bigger ω (it has a bigger v, same ω); calculator in the wrong mode.

rpm conversion isn't in 6.1 itself, but it shows up in problems, so it's included above.