Angle converter
Convert one angle into degrees, radians, gradians, turns, arc units, and a degrees-minutes-seconds reading.
Angle input
Use a single angular unit or enter a DMS value.
Required. Use a decimal point; scientific notation and decimal commas are rejected.
Minutes and seconds must be from 0 up to, but not including, 60. Put the sign on Degrees.
Converted angle
Complete conversion table
| Unit | Symbol | Converted value |
|---|
How to use the angle converter
What this calculator does
This calculator expresses one angle in several equivalent units. It is useful when a drawing uses degrees but software expects radians, when survey or coordinate data uses degrees-minutes-seconds, or when a specification uses gradians, turns, milliradians, or microradians. The result is an exact unit conversion apart from displayed rounding; it does not determine a missing geometric angle from side lengths or solve a triangle.
When to use it
Use it to prepare trigonometric inputs for programming, compare engineering or navigation specifications, translate a compass or survey reading into decimal degrees, or check common reference angles such as 30°, 45°, 90°, and 180°. The BIPM SI Brochure explains the SI treatment of plane angle and the radian.
How to calculate
- The calculator opens with a ready-to-use demonstration of 45 degrees, and the Excel workbook is immediately available.
- Choose Single value for one numeric value and select its Source unit, or choose Deg / min / sec for a compound angle.
- Replace the demonstration value. Results and the complete conversion table update live.
- Read the primary degree result, the summary cards, and the detailed unit table. Select Download Excel to export the current validated model.
- Reset clears the demonstration and all results. Excel export stays disabled until a complete valid angle is entered again.
Input guide
Input format is required and chooses between a scalar value and DMS. Angle value is a required finite decimal in the selected unit; for example, 45 with Degrees. A decimal point is accepted, while decimal commas, scientific notation, and mixed unit text are rejected to avoid silent reinterpretation. Source unit is required in Single value mode and determines the conversion factor. Higher values produce proportionally higher values in every output.
In DMS mode, Degrees is required and may carry the negative sign. Minutes and Seconds are required numbers from 0 through less than 60; for example, 48 degrees, 37 minutes, and 52 seconds. A common mistake is entering 75 minutes rather than carrying 60 minutes into one additional degree.
Output guide
Degrees is the canonical decimal-degree value. Radians uses π radians for 180 degrees. Gradians divides a full turn into 400 parts. Turns expresses the angle as a share of one revolution. π radians displays the coefficient multiplying π. DMS decomposes decimal degrees into degree, minute, and second components. Arcseconds counts 3,600 arcseconds per degree. The pills classify the angle, show its percentage of a full turn, and identify positive, negative, or zero rotation. The table also includes arcminutes, milliradians, and microradians. All are conversions, not recommendations.
Worked example
The startup value is 45°. Radians equal degrees × π ÷ 180, so 45 × π ÷ 180 = 0.7853981634 rad. Gradians equal degrees × 10 ÷ 9, giving 50 gon. Turns equal degrees ÷ 360, giving 0.125 turn. The π-radian coefficient is degrees ÷ 180, so the result is 0.25π rad. Because 45 degrees has no fractional part, its DMS form is 45° 0′ 0″, and multiplying by 3,600 gives 162,000 arcseconds.
How angle units relate
A complete revolution equals 360 degrees, 2π radians, 400 gradians, or 1 turn. Therefore every conversion can pass through a canonical degree value. The central identities are:
gradians = degrees × 10 / 9
turns = degrees / 360
The radian is especially important in calculus and programming because it relates arc length directly to radius. For additional definitions and notation, see the Wolfram MathWorld radian reference. For coordinate work, NOAA's National Geodetic Survey coordinate conversion tool provides broader geodetic transformations.
Reading negative and large angles
Negative angles usually indicate rotation in the opposite direction from the chosen positive convention. Values larger than one turn are valid and represent multiple revolutions; for example, 810° equals 2.25 turns. This converter does not automatically reduce angles to a 0° – 360° interval because preserving the original number of rotations can matter in mechanics, animation, and control systems.