Trigonometry
Degrees vs Radians
Degrees divide one revolution into 360 parts. Radians measure an angle by the ratio of arc length to radius, so one full revolution is 2 pi radians.
Reviewed
Two angle units, different strengths
Degrees divide a full turn into 360 parts. Radians compare arc length with radius, so they fit naturally into formulas and calculus.
| Measure | Full turn | Quarter turn | Best suited to |
|---|---|---|---|
| Degrees | 360 degrees | 90 degrees | Navigation, surveying, familiar angle descriptions |
| Radians | 2pi radians | pi/2 radians | Arc length, angular motion, calculus |
Two units for the same angle
An angle describes rotation, and degrees and radians assign different numbers to that rotation. A right angle is 90 degrees or pi/2 radians; neither representation changes the geometry.
Degree notation is common in surveying and everyday descriptions. Radians are natural in higher mathematics because arc length, derivatives, and periodic formulas simplify when angles are dimensionless ratios.
360 degrees = 2 pi radiansWorked example 1
Convert 225 degrees to radians
Multiply by pi/180 and reduce.
225 x pi/180Divide 225 and 180 by 45
Result: 225 degrees = 5pi/4 radians
Convert without memorizing two unrelated formulas
Multiply by a conversion factor equal to 1 and choose the form that cancels the starting unit.
1. Convert 72 degrees to radians.
Hint: Use pi radians / 180 degrees.
Answer: 72 x pi/180 = 2pi/5 radians
2. Convert 5pi/12 radians to degrees.
Hint: Use 180 degrees / pi radians.
Answer: (5pi/12) x (180/pi) = 75 degrees
Calculator mode matters
A calculator in DEG mode interprets sin(30) as the sine of 30 degrees, which is 1/2. In RAD mode it interprets 30 as 30 radians, a completely different angle.
The mode affects trigonometric functions and their inverses, not ordinary arithmetic.
Mode and notation checks
- Using pi/360 instead of pi/180.
- Entering a degree value while the calculator remains in RAD mode.
- Replacing pi with 3.14 too early and losing exactness.
Questions about degrees and radians
Are radians a unit?
The radian is the SI coherent unit for plane angle and is a ratio of two lengths, so it is dimensionless while still being named for clarity.
How many radians are in one degree?
One degree equals pi/180 radians, approximately 0.0174533.