Math101 · Independent Ontario learning library
Math101Degrees and Radians
Degrees and radians are two units for angle measure. A full turn is $360^\circ=2\pi$ radians, so $180^\circ=\pi$ radians.
Radian measure links angles directly to arc length and makes trigonometric rates and calculus formulas natural. Fluency in both units prevents calculator and formula errors.
Intuition and core definition
Degrees and radians are two units for angle measure. A full turn is $360^\circ=2\pi$ radians, so $180^\circ=\pi$ radians. One radian is the central angle intercepting an arc equal in length to the radius; this makes radian measure a natural length ratio.
Notation, language, and conditions
Convert degrees to radians by multiplying by $\pi/180^\circ$ and radians to degrees by $180^\circ/\pi$. Radian measures are dimensionless ratios but should be identified as radians. Formulas such as $s=r\theta$ and calculus derivatives of trig functions use radians.
Why this idea matters
Degrees count a turn in 360 parts, while radians compare arc length with radius and connect angle directly to circular measurement.
A dependable method
- Write the starting value with its angle unit.
- Choose a conversion factor equal to one with the desired unit on top.
- Multiply and cancel the old unit.
- Simplify the exact multiple of $\pi$ or degree fraction.
- Check against benchmark quarter, half, and full turns.
