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Interactive angle studio
Angle Converter
Convert angle units, explore an interactive unit circle, work with degrees-minutes-seconds, and translate azimuths into compass bearings.
Live diagram · DMS · navigationConvert an angle
Enter a value, choose the source and target units, or drag the arm on the diagram.
Common angles
Drag the angle arm
45°
Decimal degrees and DMS
Convert decimal degrees to degrees, minutes and seconds, or edit the DMS fields to calculate a decimal value.
Decimal degrees
DMS result—
DMS
Decimal result—
Azimuth and compass bearing
Convert a clockwise azimuth measured from north into a quadrant bearing, or build an azimuth from N/S and E/W directions.
Quadrant bearing—Nearest compass direction—
Azimuth (0–360°)—
Angle result
—
Converted value—
Normalized 0–360°—
Signed −180–180°—
Angle type—
Position—
Exact form—
Unit-circle values
Sine—
Cosine—
Tangent—
Point (cos θ, sin θ)—
The same angle in every unit
Degree—°
Radian—rad
Gradian (gon)—gon
Turn—turn
Arcminute—′
Arcsecond—″
Milliradian—mrad
Calculation steps
Angle conversion formulas
All conversions use a full turn as the shared reference. Keep extra precision during the calculation and round only the displayed result.
θrad = θdeg × π180
θdeg = θrad × 180π
Decimal degrees = degrees + minutes ÷ 60 + seconds ÷ 3600
1 turn = 360° = 2π rad = 400 gon
Where angle conversions are used
Geometry and trigonometry
Compare degree and radian measures, locate an angle by quadrant, and check sine, cosine and tangent values.
Engineering and CAD
Translate drawings, rotations and specifications while retaining enough decimal precision for technical work.
Navigation and surveying
Convert between azimuths and quadrant bearings for headings, maps and field measurements.
Astronomy and coordinates
Work with fine angular measurements using arcminutes, arcseconds and DMS notation.
Precision and interpretation
Equivalent rotations
Angles that differ by a whole turn point in the same direction. The normalized result shows the equivalent value from 0° up to, but not including, 360°.
Arcminutes are not time minutes
One arcminute is 1/60 of one degree and one arcsecond is 1/60 of one arcminute.
Tangent near 90°
Tangent is undefined when cosine is zero. Very close angles can produce extremely large finite values because of rounding.
Bearings use a different zero
Mathematical angles usually start at the positive x-axis and increase counterclockwise. Azimuths start at north and increase clockwise.
Angle Converter FAQ
How many radians are in 180 degrees?
There are π radians in 180 degrees. Therefore 1 degree equals π/180 radians.
What is the difference between degrees and gradians?
A complete turn contains 360 degrees but 400 gradians, so 1 degree equals 10/9 gradians.
How do I convert decimal degrees to DMS?
Keep the whole degrees, multiply the decimal remainder by 60 for minutes, then multiply the remaining fraction by 60 for seconds.
Can I convert negative and greater-than-360-degree angles?
Yes. The raw conversion preserves the rotation count, while the normalized cards show equivalent directions in standard ranges.
Why is tangent undefined at some angles?
Tangent equals sine divided by cosine. It is undefined at 90° plus any whole multiple of 180° because cosine is zero.
What is the difference between an azimuth and a quadrant bearing?
An azimuth is a clockwise angle from north from 0° to 360°. A quadrant bearing uses N or S, an angle up to 90°, and E or W.