Angle reference
Technical unit workspace
- Quantities
- 3
- Systems
- SI / US
Common angle units:
- Degree (°): most widely used
- Radian (rad): used in mathematics
- Gradian (grad): used in surveying
- Mil: used in military applications
Key relationships:
- 360° = 2π rad = 400 grad = 6400 mils
- 1 rad ≈ 57.2958°
- 1° = π/180 rad ≈ 0.01745 rad
Degrees:
The degree measurement originates from ancient Babylonian astronomy, which used a sexagesimal (base-60) system. The division of a circle into 360 parts may relate to the approximate number of days in a year.
Radians:
Introduced in the 18th century, the radian is considered the natural unit for measuring angles in mathematics. It represents the angle subtended by an arc whose length equals the radius of the circle.
Gradians:
Developed during the French Revolution as part of the metric system, dividing a circle into 400 equal parts (100 gradians in a right angle).
Degrees are used for:
- Navigation and mapping
- Everyday angle measurements
- Geography (latitude/longitude)
- Weather and wind direction
Radians are used for:
- Calculus and advanced mathematics
- Physics (angular velocity/momentum)
- Engineering calculations
- Signal processing
Other units:
- Gradians: surveying and engineering
- Mils: military artillery and targeting
- RPM: rotational speed in machinery
Angle Conversion Calculator
Quick answer
Formula: rad = ° x 0.0174533
Converting Degree to Radian with 6-place display precision.
Measurement type
Select a quantity to update available units
Quick pairs
Jump to the most common unit combinations
Quantity
Angle
Input unit
Degree (°)
Output unit
Radian (rad)
What are Angle Measurements?
Angle measurements quantify the amount of rotation between two intersecting lines or planes. There are several systems for measuring angles, with degrees, radians, and gradians being the most common.
The relationship between these units is precisely defined: 360 degrees (°) = 2π radians (rad) = 400 gradians (grad) = 1 revolution (rev). This relationship is fundamental to trigonometry, calculus, physics, engineering, navigation, and many other fields.
Learn more from NIST or Bureau International des Poids et Mesures (BIPM).
Common Angle Conversions
| Angle | Degrees (°) | Radians (rad) | Gradians (grad) |
|---|---|---|---|
| Right angle | 90° | π/2 rad (≈ 1.5708) | 100 grad |
| Straight angle | 180° | π rad (≈ 3.1416) | 200 grad |
| Full circle | 360° | 2π rad (≈ 6.2832) | 400 grad |
| 1 radian | 57.2958° | 1 rad | 63.662 grad |
| 30° (π/6) | 30° | π/6 rad (≈ 0.5236) | 33.333 grad |
| 45° (π/4) | 45° | π/4 rad (≈ 0.7854) | 50 grad |
| 60° (π/3) | 60° | π/3 rad (≈ 1.0472) | 66.667 grad |
Angular Velocity Conversions
| Angular Velocity | Degrees/second (°/s) | Radians/second (rad/s) | RPM |
|---|---|---|---|
| 1 RPM | 6 °/s | 0.1047 rad/s | 1 RPM |
| 1 rad/s | 57.2958 °/s | 1 rad/s | 9.5493 RPM |
| 60 RPM | 360 °/s | 6.2832 rad/s | 60 RPM |
Practical Applications of Angle Conversions
Mathematics & Education
- Converting between degrees and radians for trigonometric calculations
- Teaching students different angle measurement systems
- Solving geometric problems in different coordinate systems
Physics & Engineering
- Calculating rotational motion in physics (using radians for angular velocity)
- Engineering analysis of rotating machinery (using RPM and rad/s)
- Robotics and automation for precise angular positioning
- Optics and wave mechanics calculations
Navigation & Geography
- Maritime and aviation navigation using degrees, minutes, and seconds
- GPS coordinates and mapping systems
- Surveying and land measurement (using gradians)
Military & Defense
- Artillery targeting using military angular mil measurements
- Ballistics calculations requiring precise angle conversions
- Radar and sensor operations
Astronomy & Space
- Celestial navigation and star mapping
- Telescope positioning and tracking
- Orbital mechanics calculations
Educational Resource
This educational video from Khan Academy explains how to convert between degrees and radians, with clear examples and visual demonstrations of the mathematical relationships.
Angle Conversion Resources
Get our comprehensive angle conversion chart showing degrees, radians, gradians, and other angle units for quick reference when you're working with trigonometry and angular measurements.
Related Conversion Tools:
Angle Conversion FAQs
Find answers to common questions about angle measurements and conversions.
What are the common units for measuring angles?
The most common units for measuring angles are: 1) Degrees (°) - most widely used, ranges from 0° to 360°, 2) Radians (rad) - used in mathematics and physics, one complete circle is 2π radians, 3) Gradians (grad) - divides a circle into 400 parts, 4) Minutes and Seconds - subdivisions of degrees for precise measurements.
How do you convert between degrees and radians?
To convert between degrees and radians: 1) Degrees to Radians: multiply degrees by (π/180), 2) Radians to Degrees: multiply radians by (180/π). For example: 180° = π radians, 90° = π/2 radians, 360° = 2π radians.
Why do we use different angle measurements?
Different angle measurements serve various purposes: 1) Degrees are intuitive for everyday use and navigation, 2) Radians are preferred in calculus and physics because they simplify many mathematical formulas, 3) Gradians are sometimes used in surveying, 4) Minutes and seconds provide precise measurements for astronomy and navigation.
What is the relationship between degrees and radians?
The relationship between degrees and radians is based on the circle: 1) A complete circle is 360 degrees or 2π radians, 2) 180 degrees equals π radians, 3) 90 degrees equals π/2 radians, 4) 1 radian is approximately 57.2958 degrees. This relationship is fundamental in trigonometry and mathematics.
Angle conversion shortcuts
Common degree, radian and gradian converter links.