AP® Precalculus · Unit 3 · Topic 3.2
Radian Measure
A radian measures an angle by the arc it cuts, using the radius itself as the ruler: one radian is the angle that stretches an arc exactly as long as the radius.
Degrees chop a circle into 360 pieces for no reason tied to the circle's own geometry. Radians do the opposite: they build the unit out of the circle itself. Take the radius of a circle and lay that length along the circle's edge. The angle you sweep out doing that is one radian. Since the full circumference is 2·π times the radius, going all the way around sweeps out 2·π radians, not 360 degrees. That 2·π is not decoration, it falls straight out of the circumference formula.
Because radians come from the radius, converting between degrees and radians is just a proportion: 180 degrees equals π radians, so multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. This is why 135 degrees becomes 135·(π/180) = 3·π/4 radians: on a unit circle that angle cuts an arc of length 3·π/4, since the radius is 1.
The real payoff shows up in arc length. For any angle theta given in radians, the arc it cuts on a circle of radius r is simply s = r·theta. No conversion factor, no 360 anywhere, because theta in radians already is a ratio of arc to radius. Double the radius while keeping theta fixed and the arc length doubles right along with it. This formula is the bridge into the rest of the unit: once angles are measured this way, wrapping them around the circle to build sine and cosine as smooth, stretchable waves becomes a matter of plugging theta straight into a function, no degree-to-radian bookkeeping required.
Key terms
- radian
- the angle swept when an arc equal in length to the radius is laid along the circle's edge.
- arc length
- the distance along the curved edge of a circle between two points, given by s = r·theta for theta in radians.
- unit circle
- a circle of radius 1 centered at the origin, used so arc length and angle in radians are numerically equal.
- radian-degree conversion
- multiplying by π/180 to turn degrees into radians, or by 180/π to turn radians into degrees.
Watch it explained
- How it worksHow does one radian come from laying the radius itself along the edge of the circle?
- Worked exampleFind the radian measure of 135 degrees and show the arc it cuts on a unit circle.
- Common mistakeWhy isn't a radian just an arbitrary name for roughly 57 with no reason behind it?
- Going furtherWhat happens to arc length when the radius doubles but the angle stays put?
Check yourself
End of topic.