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Angle Converter

radians equals degrees times pi over 180

Angle Conversion =

1 Degree (°)

Solution in Other Units

UnitValue
Degree (°)1
Radian (rad)0.017453293
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Angle Conversion

Degrees divide a full rotation into 360 parts, while radians use the ratio of arc length to radius, with a full rotation equal to 2π.

radians = degrees × π/180

How It Works

This converter transforms angle values between degrees and radians. Degrees divide a full rotation into 360 parts, while radians use the ratio of arc length to radius, with a full rotation equal to 2π.

Example Problem

Convert 45 degrees to radians.

  1. Multiply by π/180: 45 × π/180.
  2. 45/180 = 0.25, so 0.25 × π = 0.7854 radians

Key Concepts

Degrees and radians are two ways to measure angles. Degrees divide a full rotation into 360 equal parts, a convention dating to ancient Babylon. Radians define an angle as the ratio of arc length to radius, making a full rotation equal to 2 pi. Most scientific and mathematical formulas use radians because trigonometric derivatives and Taylor series are simplest in radian measure.

Applications

  • Physics and engineering: converting between degrees and radians for trigonometric calculations in mechanics and wave analysis
  • Computer graphics: translating rotation angles between degree-based user interfaces and radian-based math libraries
  • Navigation: converting compass bearings (degrees) to radians for great-circle distance calculations
  • Robotics: converting joint angles between human-readable degrees and controller firmware that operates in radians

Common Mistakes

  • Passing degrees to a trig function that expects radians — most programming languages (JavaScript, Python, C) use radians by default; forgetting to convert produces wildly wrong results
  • Rounding pi to 3.14 too early — using only two decimal places introduces noticeable error in precision applications; use at least 6 significant figures
  • Confusing gradians with degrees — some calculators default to gradians (400 per full turn) instead of degrees (360 per full turn), which silently corrupts results

Frequently Asked Questions

How do you convert degrees to radians?

Multiply the degree value by π/180. For example, 90° = 90 × π/180 = π/2 ≈ 1.5708 radians.

How many radians are in a full circle?

A full circle is 2π radians, approximately 6.2832 radians, which equals 360 degrees.

When should I use radians instead of degrees?

Radians are used in calculus, physics, and most scientific formulas because they simplify mathematical expressions involving trigonometric functions.

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