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 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 radiansConversion factors
Use the equality 180 degrees = pi radians as a unit conversion. Multiply a degree value by pi/180 to obtain radians. Multiply a radian value by 180/pi to obtain degrees.
Keep pi in the exact result when possible, then calculate a decimal approximation only when the application needs it.
radians = degrees x pi/180; degrees = radians x 180/piCalculator 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.
Worked examples
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 7pi/6 radians to degrees
The pi factors cancel.
(7pi/6) x 180/pi7 x 30
Result: 7pi/6 radians = 210 degrees
Common mistakes
- 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.
Where this idea is used
- Use degrees for compass bearings and many geometry problems.
- Use radians in calculus, differential equations, and angular-frequency formulas.
- Convert explicitly when combining values from different conventions.
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.
Why do calculus formulas prefer radians?
Derivative and limit formulas take their simplest form in radians; other angle scales introduce conversion factors.