Need to convert Degrees to Radians (° → rad)? The converter above delivers instant results. Below: the exact conversion factor, a reference table, real-world examples, and the history of both units — everything you need for accurate ° to rad conversion.
The History of the Degree
The 360° circle traces to Babylon (~1500 BCE), where astronomers used a base-60 number system and divided the year into ~360 days, then divided the sky circle into 360 corresponding parts. Each degree was subdivided into 60 arcminutes and 60 arcseconds — still the basis of geographic coordinates (latitude/longitude). The 360° convention survived the metric revolution precisely because 360 has many factors (2, 3, 4, 5, 6, 8, 9, 10, 12...), making angle arithmetic unusually clean.
The History of the Radian
The radian was formalized by Roger Cotes in 1714, though the concept was implicit in Leonhard Euler's earlier work. 1 radian = the angle where arc length equals radius ≈ 57.296°. 2π radians = 360°. The radian's mathematical elegance is profound: it's the only angle unit where d(sin θ)/dθ = cos θ without a correction factor. This makes radians essential in calculus, Fourier analysis, wave mechanics, and any discipline involving periodic functions.
How to Convert Degrees to Radians
Multiply Degrees by 0.017453 to get Radians.
rad = ° × 0.017453Step-by-Step Example
- Start with your value in Degrees — for example, 25 °.
- Multiply: 25 × 0.017453 = 0.43633 rad.
Pro Tip: For quick ° to rad conversions without arithmetic, bookmark this page. The interactive tool above updates instantly as you type.
Degree to Radian Reference Table
| Degree (°) | Radian (rad) |
|---|---|
| 1 | 0.017453 rad |
| 2 | 0.034907 rad |
| 5 | 0.087266 rad |
| 10 | 0.17453 rad |
| 25 | 0.43633 rad |
| 50 | 0.87266 rad |
| 100 | 1.745329 rad |
Real-World Examples — ° to rad
- 0.017453 rad — the equivalent of 1 ° (the smallest reference point)
- 0.17453 rad — a practical mid-range value (10 °)
- 1.745329 rad — a common large-quantity reference (100 °)
Applications of Degree-to-Radian Conversion
The ° to rad conversion is needed in mathematics, navigation, surveying, and astronomy. Here's where it specifically matters:
- International specifications: Products designed in one unit system must be documented for users who work in the other.
- Scientific and technical work: Research papers and engineering drawings specify angle in a particular unit — accurate conversion prevents costly errors.
- Education: Physics, chemistry, and applied math curricula require fluency in angle unit conversion.
- Everyday tasks: Cooking, construction, fitness, and travel all involve angle conversions when switching between unit systems.
Common Mistakes When Converting Degree to Radian
- Mistake: Forgetting to switch calculator to radians mode for trigonometry
- Mistake: Confusing degrees and gradians — both use decimal notation but 90° = 100 grad
- Mistake: Using degree-based trig identities in physics formulas that expect radians
Frequently Asked Questions — Degree to Radian
How many Radians are in 1 Degree?
Exactly 0.017453 rad. The conversion factor is 0.017453 — this is derived from the precise SI definitions of both units.
What is the formula to convert Degrees to Radians?
rad = ° × 0.017453. For example: 5 ° × 0.017453 = 0.087266 rad.
How do I convert Radians back to Degrees?
Divide by 0.017453: ° = rad ÷ 0.017453. Or multiply by 57.2958.
Why would I convert Degrees to Radians?
Degrees and Radians are both used in mathematics, navigation, surveying, and astronomy, but different countries and industries prefer different units. Converting between them is necessary when reading foreign specifications, using international recipes, or collaborating across measurement systems.
Is this Degree-to-Radian converter accurate?
Yes — the factor 0.017453 is derived from official SI definitions and international standards. All calculations use full floating-point precision.
What are some real-world examples of ° to rad conversion?
1 ° = 0.017453 rad; 10 ° = 0.17453 rad; 100 ° = 1.745329 rad.
Conclusion
Converting Degrees to Radians: multiply by 0.017453; 1 ° = 0.017453 rad. The converter above handles any value instantly. Refer to the table above for quick reference values, and bookmark this page for fast ° to rad access.