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Angle Converter

Converters

Convert angles between degrees, radians, gradians, arcminutes, and arcseconds with instant results and precise calculations.. Free, private — all processing in your browser.

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The Angle Converter translates angles between all common units: degrees (most everyday use), radians (mathematics and physics), gradians (surveying), arcminutes and arcseconds (astronomy, GPS coordinates). Enter a value in any unit and see equivalents in all others instantly. Essential for mathematicians, engineers, surveyors, astronomers, and anyone who works with angles across different conventions.

Degrees divide a full rotation into 360 parts. Radians use the radius as the natural unit, with 2π radians = 360°. Gradians (also called gons) divide a rotation into 400 parts, common in surveying because 100 gons = 90°. Arcminutes are 1/60 degree, arcseconds 1/60 arcminute — used in astronomy and GPS for very fine angular measurement.

Angle Converter — key features

Five common units

Degrees, radians, gradians, arcminutes, arcseconds.

Instant conversion

Enter any value and see all other units update immediately.

Compound DMS

Degrees-minutes-seconds for GPS and astronomy.

High precision

Full floating-point precision preserved in conversions.

Pi support

Enter radians as fractions of π (e.g., \"pi/4\" for π/4).

Trigonometry helpers

See sin, cos, tan values for the entered angle.

Degree symbol

Outputs include appropriate symbols (°, rad, gon).

Client-side only

All conversion runs in your browser.

How to use the Angle Converter

  1. 1

    Choose starting unit

    Degrees, radians, gradians, arcminutes, or arcseconds.

  2. 2

    Enter value

    Type the angle value. For radians, you can use \"pi\" (e.g., pi/4 = 0.785).

  3. 3

    See all units

    Tool shows equivalent in all other units simultaneously.

  4. 4

    Copy the format you need

    Click any output to copy it.

  5. 5

    Optional — view trig values

    Check sin, cos, tan of the angle if needed.

Common use cases for the Angle Converter

Math and physics

  • →Homework: Convert between degrees and radians for trigonometry problems.
  • →Programming: JavaScript and most programming languages use radians for trig functions; convert from degrees as needed.
  • →Physics calculations: Angular velocity, oscillations, and other physics use radians.

Engineering and surveying

  • →Survey data: Convert between degrees and gradians for European survey data.
  • →CAD drawings: Different CAD tools use different units; convert as needed.
  • →Civil engineering: Construction angles, slope calculations, rotation specifications.

Astronomy and navigation

  • →GPS coordinates: Convert between decimal degrees and degrees-minutes-seconds for navigation.
  • →Telescope pointing: Arcsecond precision for astronomical targets.
  • →Celestial calculations: Hour angles, declination, and right ascension conversions.

Angle Converter — examples

Common right angles

90 degrees.

Input
90°
Output
π/2 rad, 100 gon, 5400', 324000''

Radians to degrees

Pi quarter to degrees.

Input
π/4 rad
Output
45°, 50 gon, 2700'

Decimal to DMS

GPS coordinate conversion.

Input
40.756°
Output
40°45'21.6\", 0.7114 rad, 45.28 gon

Full rotation

All units for 360°.

Input
360°
Output
2π rad, 400 gon, 21600', 1296000''

Small angle

Arcsecond example.

Input
1''
Output
0.000278°, 4.85 × 10^-6 rad — very small angle, about 1 kilometer at 207 million km distance

Technical details

Angle units and conversion factors:

Degrees (°): full rotation = 360°. Most common in everyday use, geography (latitude/longitude), and navigation.

Radians (rad): full rotation = 2π (approximately 6.283). Natural mathematical unit where arc length equals radius at 1 radian. Standard in mathematics, physics, programming (JavaScript\u2019s Math.sin and Math.cos expect radians).

Conversion: degrees × π/180 = radians. Radians × 180/π = degrees.

Gradians (grad, gon): full rotation = 400. Less common but used in surveying and some European engineering contexts because 100 gon = 90° makes right-angle calculations simple.

Conversion: degrees × 10/9 = gradians. Gradians × 9/10 = degrees.

Arcminute (arcmin, '): 1/60 degree = 0.01667°. Used in navigation, astronomy, GPS.

Arcsecond (arcsec, ''): 1/60 arcminute = 1/3600 degree. Very fine measurement for astronomy, precise GPS, surveying.

Turns/revolutions: full rotation = 1 turn. Less common but occasionally used.

Compound notation: GPS coordinates often use degrees-minutes-seconds (e.g., 40°45'36\"N). Decimal degrees (40.76°) are equivalent and easier for computer processing.

Precision considerations:
- Double-precision floating point provides about 15-17 significant digits
- Simple conversions are essentially lossless
- Round-trip through multiple unit conversions may accumulate tiny errors; usually negligible

Common conversions to know:
- 90° = π/2 rad = 100 gon = 5400' = 324000''
- 180° = π rad = 200 gon
- 360° = 2π rad = 400 gon = 1 turn

Use in code:
- JavaScript: Math.sin(Math.PI / 4) for 45° sine
- Python: math.sin(math.pi / 4) same
- Many languages have degrees-mode variants

Astronomy accuracy:
- Moon\u2019s apparent diameter: about 30 arcminutes
- Arc-second is roughly the resolution of the best professional telescopes
- GPS coordinates at arcsecond precision: about 30 meters on Earth\u2019s surface

Common problems and solutions

⚠Forgetting π in radians

Radians often written as multiples of π (π/4, π/2, π). Converting between decimal radians and π-form requires mental conversion. The tool accepts both.

⚠Degree symbol confusion

° looks similar to small o but is not. On keyboards without direct symbol, many tools accept \"deg\" or unicode ° (U+00B0).

⚠GPS format ambiguity

GPS coordinates appear in decimal degrees (40.756), DMS (40°45'21.6\"), or decimal minutes (40°45.36'). Confirm format before converting.

⚠Negative angles

Negative angles represent rotation in opposite direction. -45° = 315° (same position). Some applications require specific conventions; clarify before using.

⚠Sexagesimal rounding

Converting decimal degrees to DMS involves rounding seconds. 40.756° exactly equals 40°45'21.6\", but rounding may produce 40°45'22\" for display.

⚠Trigonometry confusion

Most programming languages’ Math.sin/cos/tan use radians. Converting degrees to radians before calling is required.

⚠Modulo behavior

360° + 90° = 450°, not 90°. For normalization, apply modulo 360. Not all tools normalize automatically.

Angle Converter — comparisons and alternatives

Compared to manual calculation, this tool instantly shows all common unit equivalents. Manual multiplication by conversion factors works but is slower for multiple conversions.

Compared to scientific calculators, this tool includes gradians, arcminutes, and arcseconds that simpler calculators skip. Scientific calculators have more math features; this tool specializes in angle units.

Compared to command-line math utilities, this tool has a friendly UI. CLI is faster for scripted use; this tool for interactive work.

Frequently asked questions about the Angle Converter

▶How many degrees is 1 radian?

1 radian equals approximately 57.2958°. Exact: 180/π degrees. Radians are the natural mathematical unit where 2π radians = full rotation.

▶How do I convert degrees to radians?

Multiply degrees by π/180. 90° × π/180 = π/2 rad (approximately 1.5708). Or use this tool which does it automatically.

▶What is a gradian?

A gradian (or gon) is 1/400 of a full rotation. 100 gon = 90° (right angle). Used in surveying because right angles are exactly 100, simplifying calculations.

▶When do I use arcminutes and arcseconds?

For fine angular measurement in astronomy, navigation, GPS coordinates, and surveying. One arcminute is 1/60°; one arcsecond is 1/3600°. Your latitude in GPS coordinates typically has arcsecond precision.

▶Does JavaScript use degrees or radians?

Radians. Math.sin(Math.PI / 4) is the sine of 45°. To use degrees, convert first: Math.sin(degrees * Math.PI / 180).

▶How do I convert GPS coordinates?

GPS coordinates come in decimal degrees (40.756°) or degrees-minutes-seconds (40°45'21.6\"). Both represent the same angle. Use this tool to convert between them for different formats.

▶Is my data private?

Yes. All conversion runs in your browser.

▶What’s the relationship between degrees and turns?

1 turn = 360° = 2π rad = 400 gon. A turn is a full rotation. Used occasionally for rotary motion descriptions.

Additional resources

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