Understanding Angle Conversions
Plane angles measure rotation or opening between two directions. This converter expresses every listed unit in degrees first, then divides by the target unit’s degree-equivalent so you can jump between radians, survey-style grads, arcminutes, NATO mils, and everyday fractions of a circle.
How Angle Conversion Works
All entries share one base: the degree (°). To convert from unit A to unit B, multiply by how many degrees one unit of A represents, then divide by how many degrees one unit of B represents.
Result = Value × (source unit in degrees) ÷ (target unit in degrees)
Example: 1 radian to degrees → 1 × (180 / π) ÷ 1 ≈ 57.296°. Two radians to a quadrant → 2 × (180 / π) ÷ 90 ≈ 1.273 quadrant.
Conversion Factors: Every Unit Explained
Each factor below is how many degrees correspond to exactly one unit of that name (the internal reference for this tool).
Degree [°]
Factor: 1 (base unit)
One three-hundred-sixtieth of a full turn in the usual sexagesimal division. Common in geography, astronomy, everyday bearings, and most school trigonometry.
Radian [rad]
Factor: 1 rad = 180 / π degrees ≈ 57.2957795131°
SI angle measure: one radian subtends an arc equal to the radius. Calculus and physics formulas often assume radians (small-angle sin θ ≈ θ in radians).
Grad [gon] and Gon
Factor: 1 grad (gon) = 0.9° (400 per full circle)
Decimal-friendly angle measure used in European surveying and some engineering contexts. Grad and gon name the same unit.
Minute [′] and second [″]
Factor: 1′ = 1/60°, 1″ = 1/3600°
Arcminutes and arcseconds subdivide degrees like clock minutes (but for angle). Used for precise celestial coordinates and geodetic latitude and longitude.
Sign
Factor: 1 sign = 30°
Twelve signs span 360°, matching the ecliptic divisions in astrology and historical astronomy.
Mil
Factor: 1 mil = 360° / 6400 ≈ 0.05625°
NATO artillery mil: 6400 mils per full circle. Other definitions (for example milliradian-based) exist; this converter follows the common 6400 convention.
Revolution [r], circle, turn
Factor: one revolution = one circle = one turn = 360°
Three names for one full rotation. Machinery, PWM, and rotation counts often quote revolutions or turns.
Quadrant and right angle
Factor: 1 quadrant = 1 right angle = 90°
A quarter turn. Navigation and geometry use “quadrant”; “right angle” matches square corners and perpendicular lines.
Sextant
Factor: 1 sextant = 60°
One sixth of a full circle. Historically tied to the optical instrument used to measure elevation angles at sea.
Common Angle Conversions at a Glance
Quick reference for frequent conversions (exact where noted).
| From | To | Formula | Example |
|---|---|---|---|
| Radians | Degrees | rad × (180 / π) | π rad = 180° |
| Degrees | Radians | ° × (π / 180) | 180° = π rad |
| Arcminutes | Degrees | ′ ÷ 60 | 60′ = 1° |
| Arcseconds | Degrees | ″ ÷ 3600 | 3600″ = 1° |
| Full turn | Degrees | turn × 360 | 0.25 turn = 90° |
| Grads | Degrees | grad × 0.9 | 100 grad = 90° |
Radians vs. Degrees in Practice
Degrees divide the circle into 360 equal parts by convention. Radians tie the angle directly to arc length on a unit circle: θ radians means an arc length of θ when the radius is 1. That is why derivatives like (sin x)′ = cos x hold when x is in radians.
Switching forms is harmless as long as you stay consistent inside a formula. Use this converter whenever you inherit mixed units (for example GPS degrees with physics radians).
Angle Conversion FAQ
How many degrees are in a radian?
One radian equals 180 / π degrees, approximately 57.2957795131°. Multiply radians by that factor to get degrees.
How many radians are in a full circle?
A full circle is 2π radians (about 6.283185). One half circle is π radians (180°).
What is a grad or gon?
Grad and gon are two names for the same unit: one four-hundredth of a full turn, or 0.9°. One right angle equals 100 grad.
How many arcminutes are in one degree?
One degree contains exactly 60 arcminutes, and one arcminute contains 60 arcseconds (so 3600 arcseconds per degree).
What does this site mean by “mil”?
Here, one mil is one 6400th of a full circle (NATO mil), about 0.05625°. Other fields use milliradians or different mil scales, so always confirm which definition your data uses.
Are revolution, circle, and turn the same?
Yes for this calculator: each means one full 360° rotation. Conversions between those labels are one-to-one.