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Math101learn.math101.caDegrees 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.
Worked example
Representations and interpretation
On a circle, degree measure partitions a turn into $360$ units, while radians compare intercepted arc length with radius. The same terminal ray can carry both labels, such as $90^\circ$ and $\pi/2$.
Reasoning about variations
A numerical angle $2$ means different rotations if interpreted as $2^\circ$ versus $2$ radians (about $114.6^\circ$). Calculator mode and formula assumptions are therefore mathematical data, not display preferences.
Common mistakes
How to check your work
- Convert back to the original unit.
- Compare with $90^\circ=\pi/2$ and $180^\circ=\pi$.
- Estimate quadrant or fraction of a turn.
Practice
- Convert $150^\circ$ to radians.
- Convert $7\pi/6$ radians to degrees.
- How many radians are in one full turn?
Answers and brief solutions
Show answers
- $\frac{5\pi}{6}$ $150\pi/180=5\pi/6$.
- $210^\circ$ Multiply by $180^\circ/\pi$.
- $2\pi$ $360^\circ=2\pi$ radians.
Synthesis and transfer
A wheel-rotation model is most direct in radians because distance equals radius times angle; converting a degree input first keeps units and scale consistent.
If a wheel turns $225^\circ$, multiplying by $\pi/180^\circ$ gives $5\pi/4$ radians. The degree unit cancels, and the result represents an arc length equal to $5\pi/4$ radii. A rough benchmark confirms the conversion: $225^\circ$ is more than a half-turn $\pi$ but less than a full turn $2\pi$. Radians make circular formulas such as $s=r\theta$ and derivatives of trigonometric functions take their simplest form. Degrees remain convenient for common subdivisions of a turn, so fluent work includes recognizing and preserving the current unit rather than treating the two numerical measures as interchangeable.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Convert $150^\circ$ to radians.
- $150\pi/180=5\pi/6$.
End of lesson
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