Degree to Radian Converter
Quickly convert from Degree to Radian.
How to convert
Formula:
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A measure of the opening between two intersecting lines or planes.
Where is it used?
In geometry, architecture, navigation (degrees), and in physics/math (radians).
Examples:
• A right angle is 90 degrees.
• A full circle is 360 degrees or 2π radians.
• 1 rad is about 57.3 degrees.
Angle conversion is fundamental in mathematics, navigation, engineering, and astronomy. Degrees, radians, gradians, and turns all describe rotations — but each system serves a different purpose and context.
An angle measures the rotation between two rays sharing a common endpoint. The most common unit is the degree (°), where a full circle = 360°. The SI-preferred unit is the radian (rad), defined as the angle subtended by an arc equal in length to the radius (full circle = 2π rad ≈ 6.2832 rad). The gradian (grad or gon) divides a right angle into 100 parts (full circle = 400 grad).
Where is it used?
- Mathematics & Physics — Trigonometry, calculus, and physics formulas use radians; most math functions in programming default to radians.
- Navigation & Surveying — Compass bearings in degrees; gradians used in some European surveying traditions.
- Engineering & CNC — Rotation, slope angles, and gear geometry in degrees or radians.
- Astronomy — Arcminutes (1/60°) and arcseconds (1/3600°) for stellar coordinates and angular diameter.
- Geography — Latitude and longitude in degrees, minutes, seconds (DMS) or decimal degrees.
Common Conversion Mistakes
Forgetting to switch calculator mode (degrees vs radians)
sin(90°) = 1, but sin(90 radians) ≈ 0.894. This is one of the most common math errors. Always check your calculator mode before computing trigonometric functions.
Confusing arcminutes with minutes of time
1 arcminute (1') = 1/60 of a degree. One minute of time = 1/60 of an hour. They share notation but measure completely different things. GPS coordinates use arcminutes; clocks use time minutes.
Assuming gradians are rarely used
Gradians (gon) are standard in European surveying and some military applications. A right angle = 100 gon makes slope calculations simpler. Many scientific calculators include a GRAD mode.
Mishandling negative angles
−90° and 270° represent the same direction but different rotations. In engineering and programming, sign conventions for clockwise vs counterclockwise rotation vary by system — always verify the convention.
Quick Reference Table
| From | To |
|---|---|
| 180° | π radians (≈ 3.14159 rad) |
| 1 radian | ≈ 57.296° |
| 90° | π/2 rad ≈ 1.5708 rad |
| 1° | 0.01745 rad |
| 360° | 2π rad ≈ 6.2832 rad |
| 1 gradian (gon) | 0.9° |
| 400 gradians | 360° |
| 1 arcminute (1') | 1/60° = 0.01667° |
Frequently Asked Questions
Why do mathematicians prefer radians over degrees?
Radians make calculus and trigonometry formulas cleaner. The derivative of sin(x) is cos(x) only when x is in radians. In degrees, you'd need a correction factor of π/180. Radians also naturally connect arc length, radius, and angle: arc length = radius × angle (in radians).
How do I convert degrees to radians?
Multiply degrees by π/180 (≈ 0.01745). Example: 45° × π/180 = π/4 ≈ 0.7854 rad. To go back, multiply radians by 180/π (≈ 57.296).
What is an arcminute and arcsecond?
1 arcminute (') = 1/60 of a degree. 1 arcsecond (") = 1/3600 of a degree. The Moon's angular diameter is about 31 arcminutes. The resolution of the human eye is about 1 arcminute. Hubble Space Telescope resolves objects to about 0.05 arcseconds.
What is a full turn in radians?
A full rotation = 2π radians ≈ 6.2832 rad. Half turn = π rad. Quarter turn = π/2 rad. These exact values appear constantly in physics, engineering, and mathematics.
Sources & Standards
- International Bureau of Weights and Measures (BIPM) — SI Brochure
- National Institute of Standards and Technology (NIST)
- ISO 80000-3:2019 Quantities and units — Space and time
Reviewed by The Unit Hub Editorial Team · March 2026