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Angle Converter

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Convert between degrees, radians, gradians, and turns instantly.

Angle Converter

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Converted value
0.785398 radians

The formula

Result = Value x (degrees per "From" unit) / (degrees per "To" unit) 1 radian = 180/pi degrees ≈ 57.2958°
Example

45 degrees converts to about 0.785398 radians, or 50 gradians, or 0.125 turns.

Step-by-step guide

  1. Enter the value you want to convert.
  2. Choose the "from" and "to" units.
  3. Read the converted result.

Where each unit is commonly used

Degrees are the everyday standard for angles, familiar from protractors and compass directions. Radians are the standard unit in most higher mathematics and physics, since many formulas (like arc length and calculus derivatives of trigonometric functions) become simpler when expressed in radians rather than degrees. Gradians (also called gons) divide a full circle into 400 parts instead of 360, historically used in some surveying and engineering contexts. Turns simply express angle as a fraction of a full rotation - half a turn is 180°, a quarter turn is 90°.

Common mistakes

Entering a value in degrees into a calculator or formula expecting radians (or vice versa) - many programming languages and scientific calculators default to radians for trigonometric functions, a very common source of errors.
Confusing gradians with degrees since both are common angle units - remember gradians divide a circle into 400 parts, not 360, so the same numeric value means a different actual angle.

Frequently asked questions

Why do radians matter so much in calculus and physics?

Radians directly relate an angle to arc length on a circle of radius 1, which makes many mathematical relationships - including derivatives and integrals of trigonometric functions - simpler and more natural than they would be using degrees.

How many degrees are in a full circle, in each unit?

A full circle is 360 degrees, 2π (about 6.2832) radians, 400 gradians, or exactly 1 turn - all four describe the exact same complete rotation.

What are arcminutes and arcseconds used for?

They subdivide a degree for extra precision - 1 degree equals 60 arcminutes, and 1 arcminute equals 60 arcseconds - commonly used in astronomy, navigation, and precise geographic coordinates.

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