Degrees split a full turn into $360$ equal parts. That number is a choice people made long ago; nothing about circles forces it. Radians measure an angle with the circle itself. You already know the arc length of a sector. Now flip the idea around: use the arc to measure the angle.
Key idea
An angle of one radian cuts off an arc exactly as long as the radius. In general an angle in radians counts how many radii fit along its arc: $$\theta = \frac{s}{r}.$$ A full turn is $2\pi$ radians.
Why it works
Draw the same angle on a small circle and a large one. The sectors are similar, so if the radius doubles, the arc doubles too. The ratio $\frac{s}{r}$ stays the same, so it describes the angle and nothing else. For a full turn the arc is the whole circumference, $2\pi r$, and $\frac{2\pi r}{r} = 2\pi \approx 6.28$. So a little more than six radii wrap around any circle.
Worked examples
Worked example: From arc and radius
A circle has radius $5$ cm. A central angle cuts off an arc of $15$ cm. What is the angle in radians?
- Count radii along the arc: $\theta = \dfrac{s}{r} = \dfrac{15}{5}$.
- $\theta = 3$ radians.
- For a feel: $3$ is a little less than $\pi \approx 3.14$, so this is just short of a half turn.
Answer: $3$ radians
Worked example: A fraction of a turn
A wheel turns a quarter of a turn. What angle is that in radians?
- A full turn is $2\pi$ radians.
- A quarter of it is $\frac{1}{4} \cdot 2\pi = \frac{\pi}{2}$.
Answer: $\frac{\pi}{2}$ radians
Worked example: From angle to arc
An angle of $2$ radians sits at the center of a circle of radius $7$ m. How long is its arc?
- Two radians means the arc is two radii long.
- $s = r\theta = 7 \cdot 2 = 14$ m.
Answer: $14$ m
Common mistake
Thinking a radian measure must contain $\pi$, so that "$2$" can't be an angle. It can: $2$ radians is the angle whose arc is two radii long, about $114.6^\circ$. The $\pi$ only shows up when the angle is a simple fraction of a full turn, like $\frac{\pi}{2}$ for a quarter turn. Quick check: compare with the landmarks $\frac{\pi}{2} \approx 1.57$, $\pi \approx 3.14$ and $\frac{3\pi}{2} \approx 4.71$ to place any radian measure in its quadrant.
A radian is a length divided by a length, so it has no unit. When an angle is written with no degree sign, like $\theta = 2$, it means radians.