Angle Units Explained: Degrees, Radians, Gradians and the Turn

A full turn is one quantity with four name tags: 360 degrees to a protractor, 2π radians to a mathematician, 400 gradians to a surveyor, and simply 1 turn to anyone who thinks in fractions of a rotation. The labels differ; the circle does not — which is why every angle conversion is a fixed ratio, not a measurement.

One circle, four scales

All four units describe how much of a full rotation an angle spans, and they interlock exactly:

1 turn = 360 degrees = 2π radians ≈ 6.28319 rad = 400 gradians
90°    = π/2 rad ≈ 1.5708      = 100 grad      = 0.25 turn
1 rad  ≈ 57.2958 degrees

Degrees slice the circle into 360 parts, radians measure the arc, gradians decimalize the right angle, and the turn sets the whole rotation to 1. Because the anchors are exact — 360, 2π, 400, 1 — converting between any pair is a multiplication by two exact ratios, never an approximation.

The radian: arc length over radius

The radian is defined by geometry rather than by slicing: lay a piece of string as long as the radius along the rim of a circle, and the angle that string subtends at the center is one radian. Since the circumference is 2π radii, a full circle is 2π radians — the reason π appears at all. One radian works out to 57.2958 degrees, and 90° is π/2. Two consequences follow. First, the radian is dimensionless (arc divided by radius), so formulas that mix angles with lengths stay clean: arc length is simply s = r·θ, and a quarter of the circumference the Circle Calculator computes — a radius-10 circle spans 62.832 around, so a 90° sector covers 15.708 of arc. Second, calculators and code both expect radians, and mode switches betray people: sin(30) in degree mode is 0.5, but in radian mode sin(30) ≈ −0.988, a wild-looking number that is mathematically correct. The no-mode-switch version of that failure inside programming languages is told in the engineering conversions guide.

Gradians and turns: the outliers that earn their keep

The gradian (also "gon") splits the right angle into 100 parts, so 90° is exactly 100 grad and a full circle is 400. It is the metric-era attempt to decimalize angles, and it survives where decimal right angles are genuinely handy: surveying, some European engineering drawings, and the GRA position on scientific calculator mode switches. The turn is the opposite philosophy — instead of dividing the circle finely, it names the whole thing 1 and lets fractions speak: a quarter turn is 90°, half a turn is 180°, three-quarters is 270°. Web CSS uses this directly — transform: rotate(0.25turn) is a quarter rotation — and animation loops, gears and robotic joints read naturally in turns, because "three and a quarter revolutions" is a sentence rather than an arithmetic problem.

Why the converter goes through the turn

Converting between any two units is easiest with one middleman, and the turn is the natural hub: divide by the source unit's exact per-turn value (degrees ÷ 360, radians ÷ 2π, gradians ÷ 400), then multiply by the target's. Two properties make this robust. Round-tripping is exact — 90° → 0.25 turn → 100 grad → 90° loses nothing beyond floating-point dust, because every ratio is a definition rather than a measured constant. And angles beyond one full rotation keep counting instead of wrapping: 405° is 1.125 turns, useful for cumulative spindle or propeller rotation, where "how far has it gone" matters more than "where does it point".

Which unit where

Degrees own the everyday: protractors, compass bearings, weather directions, and triangle work, where the three angles always sum to 180° — the check the Triangle Calculator applies when it solves a triangle from its sides. Radians own mathematics, physics and programming, wherever angles meet calculus or code. Gradians hold surveying niches. Turns own CSS, animation, and anything that speaks in revolutions. None is "right"; each is fitted to its trade — the same pattern Why Two Measurement Systems documents for length and mass.

Try it yourself

Open the Angle Converter and run three checks: 90 degrees should read exactly 100 gradians and 0.25 turns, and about 1.5708 radians; 1 radian should read 57.2958 degrees; and 405 degrees should read 1.125 turns, confirming angles count past one rotation. Then connect angle to length in the Circle Calculator using the s = r·θ idea above, and check a triangle's three angles against 180° in the Triangle Calculator.