Radians measure distance in radii

Measure a turn with the radius

Imagine taking a string exactly as long as a circle's radius. Lay it along the circle's edge. The angle swept out by that arc is one radian.

The definition is a ratio:

where $s$ is the arc length and $r$ is the radius. Equivalently, $s=r\theta$ when $\theta$ is measured in radians.

Bigger circles give the same angle

Double the circle's radius and the matching arc doubles too. Their ratio stays unchanged. A radian therefore measures the turn, independent of the circle's size.

A full circumference is $2\pi r$, so a full turn is

A half turn is $\pi$ radians, and a quarter turn is $\pi/2$. These are the same turns as $180^\circ$ and $90^\circ$; we have changed the measuring unit.

Why this unit fits the geometry

With degrees, an arc-length formula needs a conversion factor. With radians, the relationship is simply “radius times angle.” This close connection to distance is why radians are the natural unit in many formulas involving circles, sine, cosine, and rates of rotation.

Wrap the string

On a circle of radius 3, an arc has length 6. What angle does it sweep out?

Answer: $6/3=2$ radians. This is less than a half turn, since $2<\pi$.