Degree to Radian Converter
Degree / Radian / Gradian Converter
About Degree to Radian Converter
Angle Measurement Units
Angles can be measured in several units. Degrees: A full circle = 360 degrees. The most common everyday unit. Radians: A full circle = 2*pi radians. Used in calculus, physics, and engineering. One radian is the angle where arc length equals radius. Gradians (gons): A full circle = 400 gradians. Used in surveying. Turns/Revolutions: A full circle = 1 turn. Simplest conceptually.
Conversion Formulas
Degrees to Radians: radians = degrees x (pi/180). Radians to Degrees: degrees = radians x (180/pi). Degrees to Gradians: gradians = degrees x (10/9). Common conversions: 90 degrees = pi/2 radians = 100 gradians. 180 degrees = pi radians. 360 degrees = 2*pi radians. 45 degrees = pi/4 radians.
Why Radians Matter
Radians are the natural unit for angles in mathematics because they simplify formulas. Arc length = radius x angle (in radians). Derivatives of trig functions are clean: d/dx sin(x) = cos(x) only works in radians. Taylor series and limit formulas require radians. All higher mathematics and physics use radians as the default angle unit.
Frequently Asked Questions
Why are there 360 degrees in a circle?
Ancient Babylonians used base-60 counting. 360 is close to days in a year and highly divisible (by 2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180), making fractions of circles easy to express.
What exactly is one radian?
One radian is the angle at the center of a circle that subtends an arc equal in length to the radius. It equals approximately 57.2958 degrees. A full circle contains 2*pi (about 6.283) radians.
When should I use degrees vs radians?
Use degrees for everyday angles, construction, navigation, and geography. Use radians for calculus, physics, engineering formulas, and programming (most math libraries expect radians).
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