Degree to Radian Converter
Convert degrees to radians with our free online angle converter.
Quick Answer
1 Degree = 0.017453 radians
Formula: Degree × conversion factor = Radian
Use the calculator below for instant, accurate conversions.
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All conversion formulas on UnitsConverter.io have been verified against NIST (National Institute of Standards and Technology) guidelines and international SI standards. Our calculations are accurate to 10 decimal places for standard conversions and use arbitrary precision arithmetic for astronomical units.
Degree to Radian Calculator
How to Use the Degree to Radian Calculator:
- Enter the value you want to convert in the 'From' field (Degree).
- The converted value in Radian will appear automatically in the 'To' field.
- Use the dropdown menus to select different units within the Angle category.
- Click the swap button (⇌) to reverse the conversion direction.
How to Convert Degree to Radian: Step-by-Step Guide
Converting Degree to Radian involves multiplying the value by a specific conversion factor, as shown in the formula below.
Formula:
rad = ° × 0.0174533Example Calculation:
1 ° = 0.017453 rad
Disclaimer: For Reference Only
These conversion results are provided for informational purposes only. While we strive for accuracy, we make no guarantees regarding the precision of these results, especially for conversions involving extremely large or small numbers which may be subject to the inherent limitations of standard computer floating-point arithmetic.
Not for professional use. Results should be verified before use in any critical application. View our Terms of Service for more information.
Need to convert to other angle units?
View all Angle conversions →What is a Degree and a Radian?
A degree (°) divides a full circle into 360 equal parts. 1° = 1/360 of a full rotation = 60 arcminutes = 3600 arcseconds.
The degree is uniquely divisible:
- 360 = 2³ × 3² × 5
- Divisible by: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180
- This divisibility made the Babylonian base-60 system practical for navigation and astronomy
A radian is the angle subtended at the center of a circle by an arc equal in length to the radius. 1 radian ≈ 57.2958 degrees.
Key values:
- π radians = 180°
- 2π radians = 360°
- 1 radian = 180/π ≈ 57.296°
Why Radians Matter
The radian is unique because it's dimensionless:
- Definition: Arc length ÷ Radius = Angle in radians
- Dimensionless: Ratio of two lengths (meters ÷ meters = just a number)
- Natural for math: Calculus formulas work perfectly with radians (no messy constants)
- SI standard: The official unit for angles in physics and engineering
- Degrees are arbitrary: Based on Babylonian base-60 system (historical accident)
Note: The Degree is part of the imperial/US customary system, primarily used in the US, UK, and Canada for everyday measurements. The Radian belongs to the imperial/US customary system.
History of the Degree and Radian
: Why 360?
Ancient Origins (1800 BCE - Babylonia)
The Babylonians, around 1800 BCE, chose 360 degrees for several practical reasons:
- Sexagesimal System: Base-60 was their mathematical foundation (still seen in 60 seconds/minute, 60 minutes/hour)
- Solar Year Approximation: The year has roughly 365 days, and 360 was close enough for early calendars
- High Divisibility: 360 has 24 divisors—more than any smaller number except highly composite numbers
- This made navigation with compass bearings straightforward
- Allows easy division into halves (180°), quarters (90°), thirds (120°), sixths (60°), etc.
- Astronomical Observation: Ancient astronomers could divide the zodiac into 12 houses (each 30°)
Why Not 100?
The metric system later proposed 100 divisions (gradians), but degrees persisted because:
- Established Tradition: 2,000+ years of navigation, astronomy, and mathematics built on degrees
- Natural Divisibility: 360 divides more naturally than 100 for common angles
- Seamless Sub-division: 1° = 60' = 3600" (arcminutes and arcseconds) flow naturally from base-60
- Inertia in Navigation: Maritime and aviation industries deeply invested in degree-based instruments
Modern Standardization
- 1665: The degree symbol (°) officially adopted in European mathematics
- 1960s: Gradians promoted as "metric alternative" but never gained widespread adoption outside Europe
- Today: Degrees remain globally dominant; gradians used mainly in French/Swiss surveying
Common Uses and Applications: degrees vs radians
Explore the typical applications for both Degree (imperial/US) and Radian (imperial/US) to understand their common contexts.
Common Uses for degrees
Navigation: Compass bearings
- North = 0° (or 360°)
- East = 90°
- South = 180°
- West = 270°
Geometry: Measuring angles in triangles, polygons, and circles
- Acute angle: 0-90°
- Right angle: 90°
- Obtuse angle: 90-180°
- Reflex angle: 180-360°
Geography: Latitude and longitude coordinates
- Latitude: 0° (Equator) to ±90° (Poles)
- Longitude: 0° (Prime Meridian) to ±180°
Construction: Slope angles, roof pitch, ramp grades
- Wheelchair ramp: 4.76° maximum (1:12 slope)
- Standard roof pitch: 20-30° (steep roofs 45°+)
- Stairs: 30-35° typical
When to Use radians
Mathematics: Required unit for calculus (derivatives, integrals of trig functions only work with radians). Physics: Angular velocity (rad/s), rotational motion, oscillations, wave propagation. Engineering: Computer graphics, robotics, signal processing, control systems. Programming: All major languages (Python, JavaScript, C++) use radians for sin(), cos(), tan().
Additional Unit Information
About Degree (°)
How do I convert degrees to radians?
Formula: radians = degrees × (π/180) = degrees × 0.01745
Examples:
- 30° = 0.524 rad = π/6
- 45° = 0.785 rad = π/4
- 90° = 1.571 rad = π/2
- 180° = 3.142 rad = π
- 360° = 6.283 rad = 2π
How do I convert degrees to gradians?
Formula: gradians = degrees × (400/360) = degrees × 1.111
Examples:
- 90° = 100ᵍ
- 45° = 50ᵍ
- 180° = 200ᵍ
- 360° = 400ᵍ
What's the difference between arcminutes and arcseconds?
- Arcminute ('): 1/60 of a degree (used in surveying, astronomy)
- Arcsecond ("): 1/60 of an arcminute = 1/3600 of a degree (used in precise astronomy)
- Example: The Moon appears to be about 0.5° (30 arcminutes) across from Earth
Finer subdivisions exist but are rarely used outside specialized fields.
Why do we use 360 degrees instead of 100?
Historical reasons primarily:
- Ancient Babylonian sexagesimal system (base-60) was established 4,000+ years ago
- 360 is highly divisible: Can divide evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180
- 100 is less convenient: Divides nicely by 2, 4, 5, 10, 20, 25, 50—fewer options
- Maritime/Aviation Inertia: Centuries of tradition in navigation instruments
- Attempted Reform: The metric system proposed gradians (400 per circle) but never gained traction
The degree persists because it's deeply embedded in global infrastructure.
Can I use decimal degrees in navigation?
Yes, but with caveats:
- GPS Systems: Use decimal degrees by default (e.g., 40.7128°N, 74.0060°W)
- Maps: Modern digital maps support decimal degrees
- Precision Issue: Decimal notation can be confusing for precision
- 0.001° ≈ 111 meters (at equator)
- 0.00001° ≈ 1.1 meters
- Traditional Navigation: Mariners still prefer DMS (degrees/minutes/seconds) for clarity
What angle is a compass bearing of "N 30°E"?
It's 30° clockwise from north, or simply 30° on a standard compass.
- N 0°E = 0° (due North)
- N 30°E = 30° (northeast-ish)
- N 90°E = 90° (due East, but written as simply "E")
- S 30°E = 150° (southeastish)
- S 30°W = 210° (southwestish)
This notation is common in surveying and maritime contexts.
What is the latitude/longitude coordinate system?
- Latitude: Angle north (+) or south (-) of the Equator
- Equator = 0°
- North Pole = 90°N
- South Pole = 90°S
- Lines of latitude run east-west
- Longitude: Angle east (+) or west (-) of the Prime Meridian (Greenwich)
- Prime Meridian = 0°
- International Date Line ≈ 180°
- Lines of longitude run north-south
Distance Approximations:
- 1° latitude ≈ 111 km everywhere
- 1° longitude ≈ 111 km × cos(latitude)
- At equator: 111 km
- At 45°: 78.5 km
- At poles: approaches 0 km
How precise is GPS in degrees?
| Decimal Places | Precision | Application | |---|---|---| | 0 | ±111 km | Continental scale | | 1 | ±11 km | Country scale | | 2 | ±1.1 km | City scale | | 3 | ±111 meters | Building scale | | 4 | ±11 meters | Street address | | 5 | ±1.1 meters | Tree in forest | | 6 | ±0.11 meters | Surveying | | 7 | ±1.1 cm | Precision surveying | | 8 | ±1.1 mm | Scientific research |
Consumer GPS typically achieves 4-6 decimal places (11m - 1.1m accuracy).
What's the relationship between degrees and percentage grades?
- Grade (%) = tan(angle) × 100
- Examples:
- 5° slope ≈ 8.7% grade (rise 100 feet, run ~1,150 feet)
- 10° slope ≈ 17.6% grade
- 45° slope ≈ 100% grade (equal rise and run)
This is why wheelchair ramps (≤4.76° or ≤8.3% grade) are so gradual—they spread distance to minimize slope.
- 180° = 3.142 rad = π
Degrees to Radians converter →
How many degrees in a circle?
360 degrees = 1 full circle
Also:
- 90° = quarter circle (right angle)
- 180° = half circle (straight angle)
- 270° = three-quarters circle
Why 360 degrees in a circle?
Ancient Babylonians used base 60 mathematics. 360 has many divisors (2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180) making calculations easier. Also approximates 365 days in a year.
About Radian (rad)
How do I convert radians to degrees?
Formula: degrees = radians × (180/π) = radians × 57.2958
Examples:
- 1 rad = 57.3°
- π/2 rad = 90°
- π rad = 180°
- 2π rad = 360°
How many radians in 180 degrees?
180° = π radians ≈ 3.14159 radians
This is the fundamental relationship between degrees and radians.
Why use radians instead of degrees?
Mathematical simplicity: d/dx[sin(x)] = cos(x) ONLY when x is in radians. Physics advantage: Angular velocity formulas are much simpler with radians. Programming standard: All programming languages default to radians for trig functions.
Conversion Table: Degree to Radian
| Degree (°) | Radian (rad) |
|---|---|
| 0.5 | 0.009 |
| 1 | 0.018 |
| 1.5 | 0.026 |
| 2 | 0.035 |
| 5 | 0.087 |
| 10 | 0.175 |
| 25 | 0.436 |
| 50 | 0.873 |
| 100 | 1.745 |
| 250 | 4.363 |
| 500 | 8.727 |
| 1,000 | 17.453 |
People Also Ask
How do I convert Degree to Radian?
To convert Degree to Radian, enter the value in Degree in the calculator above. The conversion will happen automatically. Use our free online converter for instant and accurate results. You can also visit our angle converter page to convert between other units in this category.
Learn more →What is the conversion factor from Degree to Radian?
The conversion factor depends on the specific relationship between Degree and Radian. You can find the exact conversion formula and factor on this page. Our calculator handles all calculations automatically. See the conversion table above for common values.
Can I convert Radian back to Degree?
Yes! You can easily convert Radian back to Degree by using the swap button (⇌) in the calculator above, or by visiting our Radian to Degree converter page. You can also explore other angle conversions on our category page.
Learn more →What are common uses for Degree and Radian?
Degree and Radian are both standard units used in angle measurements. They are commonly used in various applications including engineering, construction, cooking, and scientific research. Browse our angle converter for more conversion options.
For more angle conversion questions, visit our FAQ page or explore our conversion guides.
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📚 How to Convert Units
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🔢 Conversion Formulas
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⚖️ Metric vs Imperial
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⚠️ Common Mistakes
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Verified Against Authority Standards
All conversion formulas have been verified against international standards and authoritative sources to ensure maximum accuracy and reliability.
National Institute of Standards and Technology — Fundamental physical constants and unit conversions
Bureau International des Poids et Mesures — International System of Units (SI) definitions and standards
International Organization for Standardization — Quantities and units international standards series
Last verified: February 19, 2026