Radians and arc length

Parametric equations

On a circle of radius $r$ , the arc subtended by a central angle $θ$ in radians has length

s = r θ

On the unit circle ( $r = 1$ ), the radian measure and the arc length are the same number. Degrees convert to radians by

θ_{rad} = θ_{deg} \cdot \frac{π}{18 0 ^{\circ}}

Angle θ: drag the point on the circle or the slider to change $θ$ ; the arc length $s$ , the radian value, and the degree value update together
A colored arc marks the arc from $(r, 0)$ to $P (θ)$ . Its length is $s = r θ$ . On the unit circle that length equals the radian measure.
r = 1 and r = 2: switch the radius and watch the arc length scale with the radius at the same angle
Special angles: on: mark $π /6$ , $π /4$ , $π /3$ , $π /2$ , and $π$ on the circle

The radian measure ties the size of an angle to arc length. On the unit circle, reading the radian measure is reading the arc length.
Double the radius and the arc length doubles. The angle itself, in radians, does not change with the size of the circle.
A negative angle traces the arc the other way, and the arc length is still $r ∣ θ ∣$ .