📐 Angle

Angle Converter

Degrees for everyday geometry, radians for calculus, gradians for surveying — convert any plane angle. Enter a value and press Convert.

Written and reviewed by SM Arijul Islam · Last reviewed 28 October 2026 · Editorial standards · Report an error

About the angle converter

A full circle contains 360 degrees, 2π radians, 400 gradians or one turn. Radians are the natural unit of mathematics because they relate an arc length directly to a radius, while degrees remain convenient for everyday directions and navigation.

Reference conversions

180° = π rad ≈ 3.14159265 rad
90° = 1.57079633 rad = 100 gradians = 0.25 turn
1 radian ≈ 57.29577951°
1 degree = 60 arcminutes = 3,600 arcseconds
1 milliradian ≈ 3.43775 arcminutes

Trigonometry in the site's Scientific Calculator respects the DEG/RAD switch in the same way.

How to convert angles

An angle measures rotation around a point. The degree divides a full turn into 360 equal parts, while the radian — the SI unit — is defined by the arc it subtends: a full turn is exactly 2π radians. To convert degrees to radians, multiply by π/180 (about 0.017453293); to convert back, multiply by 180/π (about 57.295780). Gradians split a right angle into 100 parts, and arcminutes and arcseconds subdivide one degree into 60 and 3,600 parts respectively.

Common angles conversions

One degree expressed in other units of angles
Unit	1 degree equals
Arcminute (')	60
Milliradian (mrad)	17.453293
Gradian (grad)	1.1111111
Radian (rad)	0.017453293
Turn (revolution) (turn)	0.0027777778

Everyday uses

Reading GPS bearings and compass headings
Geometry and trigonometry homework
Setting servo and robot joint angles
Astronomy coordinates and sky charts
Surveying and land measurement

Frequently asked questions

How many radians are in a full circle?

2π, which is about 6.2831853 rad. A right angle is π/2, or about 1.5707963 rad.

Why are there 360 degrees in a circle?

It is a convention inherited from ancient Babylonian astronomy, whose base-60 number system divides 360 in many convenient ways.

What is a gradian?

A unit that splits a right angle into 100 parts, so a full turn is 400 gradians. Some surveying instruments and calculators use it.

Are arcminutes the same as minutes of time?

No. An arcminute is one sixtieth of a degree of angle; a minute of time is one sixtieth of an hour. The names are shared, the quantities are not.

How many degrees are in one turn?

One full turn is 360 degrees, 2π radians, or 400 gradians. A half turn is 180 degrees and a quarter turn is 90 degrees.

More worked examples

Half a turn is 180°, which is pi radians, or about 3.14159 rad.
A right angle of 90° is pi/2, about 1.5708 rad.
One radian is 180 ÷ pi, about 57.2958°, so a radian is a little under 60°.
A 45° mitre cut is 0.785398 rad.

Degrees to radians

Common angles in degrees and the equivalent in radians
Degrees	Radians
30°	0.5236 rad
45°	0.7854 rad
60°	1.0472 rad
90°	1.5708 rad
180°	3.1416 rad
270°	4.7124 rad
360°	6.2832 rad

Common mistakes to avoid

Feeding degrees to a function that expects radians. In most programming languages sin, cos and tan take radians.
Using 3.14 for pi in a chain of calculations; the error compounds quickly. Use 3.14159265 or a built-in constant.
Thinking a full turn is pi radians. It is 2 pi radians, because the circumference is 2 pi times the radius.
Mixing up milliradians and degrees. One milliradian is only about 0.0573°, so 1,000 mrad makes one radian.

Further questions

Why does a full circle contain 2 pi radians?

A radian is the angle you get when the arc length equals the radius. The whole circumference is 2 pi radii long, so a full turn is 2 pi radians.

How do I convert radians to degrees without a calculator?

Multiply by 180 and divide by pi. For a rough check, remember that one radian is a little under 60°.

What is a milliradian used for?

Optical sights, artillery and surveying. One milliradian subtends about one metre at a distance of 1,000 metres.

Are there other angle units besides degrees and radians?

Yes. Surveyors use the gradian or gon, where 400 gon makes a full turn, and navigators use the compass point and the turn.

Sources and standards

The radian is the SI unit of plane angle, defined as the ratio of arc length to radius, so a full turn is exactly 2π radians. The degree is 1/360 of a turn and the gradian 1/400; both are definitions, not measurements.