Convert degrees, radians, gradians, and turns with exact factors. Free with steps.
Angles are measured in four competing units: degrees on every protractor, radians in every calculus class, gradians on some survey instruments, and turns wherever rotations count. Our free Angle Converter translates between all four with full-precision factors, showing the degree intermediate so each conversion is checkable.
The radian is the mathematician’s native unit because it makes calculus clean: the derivative of sin(x) is cos(x) only when x is in radians. The degree is the human unit: 360 slices with ancient astronomical roots. The gradian is the metric dream that never quite woke up: 400 gradians per circle. Turns are the engineer’s shortcut: 1.5 turns needs no π at all.
Convert before you compute. Trigonometric functions in code and calculators default to radians; feeding them degrees is the classic silent error — sin(30) is −0.988 in radians but 0.5 in degrees.
The sections below build full fluency: worked conversions for the famous anchors (π radians, one radian in degrees), practical stories from machine shops and survey crews, and a FAQ covering the mode-mismatch errors that trap even experienced engineers. By the end, switching between degrees, radians, gradians, and turns will feel as natural as switching between inches and feet — and your trigonometry will finally stop surprising you.
A degree splits a circle into 360 parts — a legacy of Babylonian base-60 astronomy, since 360 ≈ the days in a year and divides by 2, 3, 4, 5, 6, 8, 9, 10, and 12. Every compass, protractor, and latitude line speaks degrees.
A radian is the angle subtended by an arc equal in length to the radius: a full circle is 2π radians (≈ 6.2832). It is the SI-ish natural unit because arc length = radius × angle-in-radians, with no conversion constant — the formula s = rθ only works in radians, which is why physics and calculus demand them.
A gradian (gon) divides the circle into 400: 100 gradians per right angle, the French Revolution’s metric angle. It survives in European surveying and some calculators’ DRG mode. A turn (revolution) is simply one full circle — 360°, 2π rad — the unit of motors, gears, and coils. Working in turns keeps multi-rotation problems beautifully simple: a shaft at 3.5 turns per second needs no π to describe, and gear ratios become plain fractions.
Key terms:
Memorize the anchors: 180° = π rad, 90° = 100 gon, 360° = 1 turn. Every other conversion is a proportion from these.
Convert a straight angle to radians.
The most famous conversion in mathematics: half a circle is π radians, exactly.
How many degrees is one radian?
One radian is a surprisingly large angle — over 57 degrees — which is why small-angle approximations (θ ≈ sin θ) work so well below ~0.3 rad.
A machinist programs a rotary table: the print calls for a bolt circle with 8 holes, and the controller thinks in degrees while her CAD sketch labeled the step as 0.7854 radians. Convert: 0.7854 × 180 ÷ π = 45.0006° ≈ 45° — exactly one-eighth of a turn, as expected for 8 holes. She programs 45° steps with confidence instead of trusting a rounded radian readout.
The reverse trip bites in robotics: a servo datasheet quotes 300°/s slew rate, but the control loop’s kinematics run in rad/s. 300 × π ÷ 180 = 5.236 rad/s. Feeding 300 into a radians-expecting controller would spin the joint 57 times too fast — the classic unit mismatch that shreds gearboxes.
A second scenario: land survey. A European total station reads 97.5432 gon; the American crew’s plats are in degrees: 97.5432 × 0.9 = 87.7889°. And the DMS crew wants degrees-minutes-seconds: 0.7889° × 60 = 47.33′, so 87°47′20″. Three notations, one angle, zero ambiguity.
The 360-degree circle is Babylonian: base-60 arithmetic made 360 (6 × 60) the natural circle number, and it approximated the year’s ~360 days closely enough for ancient astronomy. Its divisibility — 24 divisors — made it immortal: navigators, clockmakers, and protractor printers never found a reason to leave.
The radian arrived with 18th-century analysis. Euler’s formula e^{iπ} + 1 = 0 fuses radians with exponentials so deeply that “angle” in higher mathematics is really a pure number — arc divided by radius, units canceling out. Radians are not just convenient for calculus; they are what angles are once geometry meets analysis.
The gradian was the metric system’s 1790s moonshot: 400 grades per circle, 100 per quadrant, decimal subdivisions everywhere. It failed because 360’s divisibility beat 400’s decimality — thirds of a right angle are 33.33… gon but a clean 30°. France’s artillery kept it; the world kept degrees.
The turn is having a quiet renaissance via the τ (tau) movement: 1 turn = τ radians (τ = 2π). “A quarter turn is τ/4” reads more naturally than “π/2,” and educators argue it repairs students’ circle intuition. Whether τ wins or not, thinking in turns — fractions of a whole rotation — is the most human angle unit of all.
Navigation exposes a fifth dialect: the compass point. Mariners once divided the circle into 32 points (11.25° each) — “two points off the starboard bow” — and modern bearings still use 0–360° clockwise from north, unlike mathematics’ counterclockwise-from-east convention. Converting a math angle to a compass bearing requires both a unit check and a reference-frame flip: bearing = 90° − math-angle (mod 360). The converter handles the units; the navigator must handle the orientation.
Astronomers add arcminutes and arcseconds for tiny angles: the Moon spans about 31 arcminutes (0.52°), and stellar parallax is measured in milliarcseconds. These are subdivisions of the degree, not separate units, but converting 0.0003° into 1.08 arcseconds makes the number human-readable. Whenever an angle shrinks below a degree, the sexagesimal subdivisions take over — the same base-60 instinct that gave us 360 degrees in the first place.
The steradian extends the story into three dimensions: it is the “square radian,” measuring solid angles, with 4π steradians in a full sphere. Antenna beamwidths, camera fields of view, and radiative lightbulbs are specified in steradians, and converting a planar angle to a solid angle (for small angles, Ω ≈ θ²) is where 2D angle fluency graduates into 3D. Degrees got us around the circle; steradians get us around the sky.
Angle conversions: rad = deg×π/180; deg = rad×180/π; gon = deg×10/9; turns = deg/360. Anchors: 180°=π rad, 90°=100 gon, 360°=1 turn. 1 rad ≈ 57.2958°. Trig functions need matching calculator mode (DEG/RAD/GRD). Source: Sigma Calculator, Angle Converter, https://sigmacalculator.net/math-calculators/angle-converter/
Cite this page as: Angle Converter, Sigma Calculator, https://sigmacalculator.net/math-calculators/angle-converter/
Last updated: October 12, 2026