Angle Unit Converter - Degrees, Radians, Gradians

Four ways to measure the same angle

Type an angle in any of the four supported units and the converter shows all the others - plus DMS notation and trigonometric values. It also accepts direct DMS input (degrees, minutes, seconds) when you select degrees as the input unit.

360

degrees in a full circle

2π

radians in a full circle

400

gradians in a full circle

turn = one full rotation

Unit comparison table

Unit	Full circle	Right angle	Used in
Degrees	360	90	Everyday use, navigation, construction
Radians	2π (6.2832)	π/2 (1.5708)	Mathematics, physics, programming
Gradians	400	100	Surveying, some European countries
Turns	1	0.25	Rotation speed (RPM), motors

Key conversion formulas

Conversion	Formula	Example
Degrees to radians	deg x π/180	90 x π/180 = π/2
Radians to degrees	rad x 180/π	π x 180/π = 180
Degrees to gradians	deg / 0.9	90 / 0.9 = 100 grad
Degrees to turns	deg / 360	180 / 360 = 0.5 turns

Practical examples

Right angle: 90 degrees
= π/2 rad = 100 grad = 0.25 turns | sin=1, cos=0, tan=undefined

45-degree angle: 45 degrees
= π/4 rad = 50 grad = 0.125 turns | sin=cos=0.7071

Programming: JavaScript Math.sin() takes radians
Math.sin(90) is wrong - you need Math.sin(90 * Math.PI / 180) = Math.sin(1.5708)

Surveying with gradians: A right angle is exactly 100 grad
Makes calculation easier: half a right angle = 50 grad, quarter = 25 grad

DMS notation: GPS coordinate 51 degrees 30 minutes 26 seconds
= 51 + 30/60 + 26/3600 = 51.5072 decimal degrees

Motor speed: 1500 RPM = 1500 turns per minute
= 540,000 degrees/min = 9424.78 rad/s

Why 360 degrees?

The Babylonians used base-60 arithmetic, and 360 is close to the number of days in a year. A circle divided into 360 parts gives a unit small enough for practical measurement yet large enough to avoid fractions in common angles. The number 360 has 24 divisors, so it splits evenly into halves (180), thirds (120), quarters (90), fifths (72), sixths (60), eighths (45), tenths (36), twelfths (30) and more. No other small number is this divisible.

Radians, by contrast, are defined by the geometry of the circle itself: one radian is the angle subtended by an arc equal in length to the radius. This makes them the natural unit for calculus and physics, where formulas like arc length = r x theta only work when theta is in radians.

FAQ

What is the difference between degrees and radians?

Both measure angles, but use different scales. Degrees divide a circle into 360 equal parts - intuitive for everyday use. Radians relate the angle to the radius of the circle: an angle of 1 radian covers an arc equal to the radius. A full circle is 2π radians (about 6.283). Most programming languages and scientific formulas expect radians.

Who uses gradians?

Surveyors and civil engineers, especially in France, Germany, and other parts of continental Europe. Gradians divide a right angle into exactly 100 parts, making slope and bearing calculations easier with decimals. Most scientific calculators have a DEG/RAD/GRAD mode switch.

What is DMS notation?

Degrees, Minutes, Seconds - a way to express fractional degrees without decimals. One degree has 60 minutes ('), one minute has 60 seconds ("). Example: 45.5072 degrees = 45 degrees 30 minutes 15.79 seconds. GPS coordinates are often given in DMS format.

Can I enter negative angles?

Yes. Negative angles represent clockwise rotation (positive is counter-clockwise by convention). The converter handles angles from -999999 to 999999 in any unit. A -90 degree angle has the same magnitude as +90 degrees but points in the opposite direction.

Why does the calculator show sin, cos and tan?

Trigonometric functions are the most common reason to convert angles. If you enter 30 degrees, the converter shows sin(30)=0.5, cos(30)=0.866, tan(30)=0.577 - saving you a separate calculation. The values are computed from the radian equivalent regardless of which input unit you used.

When is tan undefined?

Tangent is undefined at 90 degrees (π/2 rad) and 270 degrees (3π/2 rad) - essentially any odd multiple of 90 degrees. At these angles the cosine is zero, and tan = sin/cos would mean dividing by zero. The converter displays "undefined" instead of a number.

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