🔺 Angle Converter
Convert between degrees, radians, gradians, arcminutes, arcseconds, and full turns for geometry, navigation, and engineering.
Enter a value above to see the result
Common Angle Conversions
| Input | Unit | Result | Unit |
|---|---|---|---|
| 180 | Degree | 3.1415927 | Radian |
| 1 | Radian | 57.29578 | Degree |
| 90 | Degree | 100 | Gradian |
| 1 | Degree | 60 | Arcminute |
| 1 | Full Turn | 360 | Degree |
| 3600 | Arcsecond | 1 | Degree |
Popular Conversions
Formula Reference
1 full turn = 360° = 2π rad ≈ 6.2832 rad = 400 grad 1° = 60 arcminutes (′) = 3,600 arcseconds (″) rad → deg: multiply by 180/π ≈ 57.2958 deg → rad: multiply by π/180 ≈ 0.017453
About Angle Conversions
Angles can be measured in several systems depending on context. Degrees are the most familiar: a full circle is 360°, a right angle is 90°, and a straight line is 180°. The choice of 360 is ancient, rooted in Babylonian mathematics and the approximate number of days in a year. Most people encounter degrees in everyday geometry, navigation, and weather (wind direction, bearing).
Radians are the mathematically natural unit for angles and the standard in all scientific and engineering calculations. A radian is the angle subtended at the center of a circle by an arc equal in length to the radius. Since the full circumference is 2π radii, a full circle is exactly 2π radians. Radians make calculus formulas for sine and cosine particularly clean — derivatives of sin(x) and cos(x) in radians are exactly cos(x) and −sin(x), without any scaling factor.
Gradians (also called gon or grade) divide a full circle into 400 units, placing a right angle at exactly 100 gradians. This was introduced during the French Revolution alongside the metric system. While gradians never achieved widespread adoption, they remain in use in surveying and civil engineering in parts of Europe. Arcminutes and arcseconds subdivide degrees (1° = 60′ = 3,600″) and are essential in astronomy, GPS coordinates, and optical precision measurement.
Frequently Asked Questions
How do I convert degrees to radians?
Multiply the degree value by π/180 (approximately 0.017453). Common conversions: 30° = π/6 ≈ 0.5236 rad, 45° = π/4 ≈ 0.7854 rad, 90° = π/2 ≈ 1.5708 rad, 180° = π ≈ 3.1416 rad, 360° = 2π ≈ 6.2832 rad. Most scientific calculators have a dedicated degree/radian mode switch.
Why do mathematicians prefer radians?
Radians arise naturally from the geometry of circles and make calculus formulas cleaner. The derivative of sin(x) is cos(x) only when x is in radians — in degrees, a correction factor of π/180 is needed everywhere. Power series for sine and cosine, Euler's formula (e^(iθ) = cos θ + i sin θ), and arc length formulas all assume radians. Degrees are a historical convention; radians are the mathematically fundamental unit.
What is an arcminute and when is it used?
One arcminute (′) equals 1/60 of a degree. One arcsecond (″) equals 1/60 of an arcminute, or 1/3,600 of a degree. These are used in astronomy (angular diameters of stars and planets), GPS coordinates (latitude/longitude are expressed in degrees, minutes, and seconds), and precision optics. The resolution of the human eye is about 1 arcminute — meaning two points closer than 1′ apart at a given distance appear as one.
What is a gradian?
A gradian (also gon or grade) divides a full circle into 400 equal parts, placing a right angle at exactly 100 gradians. It was introduced in France during the Revolution to align with the metric system's base-10 philosophy. While it never replaced degrees broadly, gradians are still used in surveying, civil engineering, and some European countries' construction standards.
What does 'subtended angle' mean?
A subtended angle is the angle formed at a point when two lines extend from that point to the endpoints of an arc or segment. The radian is defined as the angle subtended at the center of a circle by an arc whose length equals the radius. This geometric definition is why radians are dimensionless (they are a ratio of two lengths) and why they simplify so many mathematical formulas.
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All conversion results are provided for general informational purposes only. Our formulas use internationally recognized conversion factors; however, rounding may affect precision. Do not rely on these results for professional, medical, legal, or engineering decisions without independent verification. Read our full disclaimer.