Radians & the Unit Circle

An angle is how much arc it cuts: 180^ = π radians.

The idea

A radian measures an angle by the circle itself, where a degree is only a convention: someone cut a full turn into $360$ parts, and nothing about a circle proposes that number.

Open an angle $\theta$ at the center $O$ of a circle of radius $r$, and it cuts off an arc of length $s$. Enlarging the circle grows $s$ and $r$ together, so the ratio $s/r$ depends on the angle alone.

Definition (Radian).

The radian measure of a central angle is the ratio $\theta = s/r$ of the arc it cuts off to the radius: the angle is $\theta$ radians when its arc is $\theta$ times as long as the radius. One radian is the angle whose arc is exactly one radius long.

A full turn cuts off the whole circumference $2\pi r$, which is $2\pi$ radii of arc, so a full turn is $2\pi$ radians and a half turn is $180^{\circ} = \pi \text{ radians}.$ Every conversion comes from this equation: multiply degrees by $\pi/180$, or radians by $180/\pi$.

The definition also rearranges into the arc-length formula $s = r\theta$, with no correction factor — in degrees the same formula needs one. On the unit circle, the circle of radius $1$, the angle and its arc are the same number.

Ways to work on it

• Walkthrough. What a radian is, converting both ways, and why s = rθ is the payoff.
• Practice. Convert common angles between degrees and radians.
• Hardest. Find an arc length from an angle measured in degrees.

Not sure where to start? Take the ten-question placement test.