Convert Degrees to Radians - Angle Converter
Convert angles from degrees to radians using the exact mathematical relationship where π radians equals 180 degrees. This converter returns exact symbolic results when possible and high-precision numeric values for engineering and scientific use.
Radians are the standard angular unit in calculus, physics, and many engineering disciplines because they relate angle measure directly to arc length and derivatives. Use radians when working with trigonometric functions, differential equations, or any formula that assumes SI-consistent units.
This tool includes practical notes on numeric precision, common pitfalls (negative angles, wrap-around, and degrees-minutes-seconds), and guidance for interpreting results with instrument and software limitations.
Interactive Converter
Convert between degree and radian with precision rounding.
Quick reference table
| Degree | Radian |
|---|---|
| 1 ° | 0.0175 rad |
| 5 ° | 0.0873 rad |
| 10 ° | 0.1745 rad |
| 25 ° | 0.4363 rad |
| 50 ° | 0.8727 rad |
| 100 ° | 1.7453 rad |
Methodology
The conversion is based on the mathematical identity: 1 degree = π / 180 radians. This identity is consistent with SI conventions for plane angle (radian as the coherent derived unit) as described by international measurement authorities.
For exact values, angles that are rational multiples of 180° can be expressed as rational multiples of π (for example, 45° = π/4). For numeric output, the converter uses full double-precision arithmetic and provides guidelines for rounding to suit lab, classroom, or production requirements.
Practical notes: floating-point results are subject to IEEE-754 rounding. For instrumentation or regulatory reporting, follow your lab’s calibration and uncertainty procedures and, when required, present results with documented precision and units.
Worked examples
45 degrees → π/4 radians → approximately 0.7853981633974483
180 degrees → π radians → exactly π (approximately 3.141592653589793)
1 degree → π/180 radians → approximately 0.017453292519943295
Further resources
Expert Q&A
What is the exact relationship between degrees and radians?
1 degree = π / 180 radians. Equivalently, 1 radian = 180 / π degrees. This is the standard mathematical definition used in calculus and SI-consistent measurements.
Should I use degrees or radians in calculus and physics?
Use radians for calculus and most physics formulas because derivatives and series expansions (for example, d/dx sin(x) = cos(x)) assume x is in radians. Degrees are often used for human-friendly display, navigation, or some engineering contexts, but convert to radians for computation.
How do I handle negative angles and wrap-around?
Apply the same conversion formula; sign is preserved. For wrap-around into a principal interval, reduce modulo 2π after converting (for radians) or modulo 360° before converting, depending on your workflow.
How accurate are numeric conversions here?
Numeric results use double-precision evaluation of π and standard IEEE-754 arithmetic. For most lab and engineering uses this gives accuracy well below typical measurement uncertainty. If you require stated uncertainty, apply your instrument calibration and uncertainty propagation procedures.
Can I convert degrees-minutes-seconds (DMS) to radians?
Yes. Convert DMS to decimal degrees first (degrees + minutes/60 + seconds/3600), then multiply by π/180 to get radians. Be careful to preserve sign for southern/western or negative angles.
Are radians a base SI unit?
The radian is the SI coherent derived unit for plane angle and is dimensionless in the SI system; its use and definition are described by international metrology authorities. For authoritative details, consult SI documentation from national metrology institutes.
How should I present converted values for reports or regulatory submissions?
Report numeric values with appropriate significant figures and units. Include the conversion formula or reference, the precision of the numeric evaluation, and any uncertainty from measurement or rounding. Follow your organization's reporting standards and any applicable regulatory guidance.
How can I get symbolic results (in terms of π) instead of decimal approximations?
Angles that are rational multiples of 180° can be expressed exactly as fractions of π (for example, 60° = π/3). When an exact symbolic form exists, present it alongside the decimal approximation for clarity.
Sources & citations
- BIPM — SI derived units and the radian — https://www.bipm.org/en/measurement-units/
- NIST — The International System of Units (SI) and plane angle — https://www.nist.gov/pml/special-publication-811
- MIT OpenCourseWare — Single Variable Calculus (definitions and use of radians) — https://ocw.mit.edu/courses/18-01-single-variable-calculus/