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Angle Converter

Convert degrees, radians, gradians, arcminutes, and arcseconds for trig and surveying prep.

About Angle Converter

Angles appear in four different unit systems depending on the field: degrees for everyday geometry, radians for calculus and programming, gradians for European surveying, and turns for animation and robotics. Converting between them manually is a source of constant errors — the degree-to-radian step trips up every programmer who forgets that JavaScript trig functions expect radians. This converter handles all four units simultaneously: degrees (°), radians (rad), gradians (grad), and turns. It also supports degrees-minutes-seconds (DMS) notation used in GPS coordinates and navigation charts. Type in any field and all others update instantly. An optional π-fraction display shows radians as multiples of π (π/2, π/4) for textbook-style communication. A wrap-around normalisation option constrains results to 0–360° or −π to π for animation and rotation code.

Why use Angle Converter

Four Units

Converts between degrees, radians, gradians, and turns — all four units in common professional and academic use — in a single interface. No mental formula juggling or unit-specific tool switching required.

DMS Notation

Supports degrees-minutes-seconds input and output for GPS coordinates, navigation charts, and surveying data where decimal degrees are not standard. Enter 37° 25′ 30″ and read the decimal and radian equivalents immediately.

Live Multi-unit

All four angle unit fields update on every keystroke. Watching 90° simultaneously read as π/2 rad, 100 grad, and 0.25 turn makes the relationships between units concrete and intuitive without any calculation effort.

π-form Output

Optional display of radian values as π-fraction notation (π/2, π/4, 2π/3). This format is standard in textbooks, exam solutions, and code documentation where decimal approximations (1.5708) are less communicative.

Wrap-around Safe

Normalise any angle to the 0–360° range or the −π to π range with one toggle. Useful for animation code, rotation math, and navigation calculations where angles outside the standard range cause unexpected behaviour.

Offline Capable

All trigonometric and angular conversions run as pure browser JavaScript. No network request is needed, so the tool works in field conditions, on planes, or in labs where internet access is restricted.

How to use Angle Converter

  1. Select the source unit: degrees, radians, gradians, turns, or DMS (degrees-minutes-seconds).
  2. Enter your angle value in the input field for your chosen source unit.
  3. Read all other units update simultaneously in the output fields.
  4. Toggle the π-fraction display to see radian values expressed as multiples of π.
  5. Use the wrap-around option if you need the result normalised to 0–360° or −π to π.
  6. Copy any output field with its unit label for use in code, coordinates, or reports.

When to use Angle Converter

  • Converting degrees to radians before passing an angle to JavaScript's Math.sin() or Math.cos() functions.
  • Translating GPS or survey DMS coordinates (37° 25′ 30″) to decimal degrees for a mapping API.
  • Understanding a gradian value in a civil engineering or European surveying document.
  • Expressing a radian value as a π-fraction for a calculus exam solution or textbook.
  • Normalising rotation angles to the 0–360° range in an animation or game engine.
  • Verifying that a full rotation (360°) equals exactly 2π rad, 400 grad, and 1 turn.

Examples

Right angle

Input: 90°

Output: π/2 rad = 1.5708 rad = 100 grad = 0.25 turn

DMS to decimal

Input: 37° 25′ 30″

Output: 37.4250° = 0.6535 rad

Full circle

Input: 360°

Output: 2π rad = 6.2832 rad = 400 grad = 1 turn

Tips

  • JavaScript's Math.sin(), Math.cos(), and Math.atan2() all use radians — always convert from degrees before passing values into these functions.
  • DMS notation uses minutes (1/60 of a degree) and seconds (1/3600 of a degree) — exactly like time notation but applied to angles.
  • Gradians make right angles a clean 100 grad — useful in some surveying contexts where 90° produces less convenient arithmetic.
  • One turn = 360° = 2π rad — this is the cleanest way to think about full rotations in animation and robotics code.
  • Always verify the quadrant (I, II, III, IV) when normalising negative angles — wrapping can move a value to the wrong quadrant if the sign is not handled carefully.

Frequently Asked Questions

What is the relationship between degrees and radians?▾
180° = π radians exactly. To convert degrees to radians, multiply by π/180 (≈ 0.01745). To convert radians to degrees, multiply by 180/π (≈ 57.296). This relationship arises because the circumference of a unit circle is 2π.
How many radians are in a full circle?▾
Exactly 2π radians, approximately 6.28318 rad. This equals 360° or 400 gradians or 1 full turn. The 2π value comes from the fact that the circumference of a unit-radius circle is 2π times the radius.
What is a gradian and where is it used?▾
A gradian (also called gon) divides a right angle into 100 units, so a full circle is 400 gradians. It was promoted during the French Revolution alongside the metric system and is still used in some European land surveying and civil engineering.
How do I convert decimal degrees to DMS?▾
Take the whole number as degrees. Multiply the decimal fraction by 60 — the whole-number part of that result is minutes. Multiply the remaining decimal by 60 again for seconds. For example, 37.425° = 37° 25′ 30″.
Why do programming languages use radians by default?▾
Radians are the natural angular unit for calculus — derivatives of trigonometric functions are only clean (d/dx sin x = cos x) when x is in radians. Using degrees would require correction factors throughout calculus-based libraries.
What is a turn?▾
One turn is one full rotation, equal to 360°, 2π radians, or 400 gradians. The turn is increasingly used in computer graphics, animation, and robotics because it maps directly to the concept of one complete revolution without requiring π.
How accurate are GPS coordinates in DMS?▾
GPS coordinates at 1-arcsecond resolution (1/3600°) correspond to roughly 30 metres on the ground at the equator. Consumer GPS devices report to about 0.0001° decimal precision, equivalent to about 11 metres.
When should I use radians over degrees?▾
Use radians for any mathematical computation involving trigonometric functions, calculus, physics formulas, or programming. Use degrees for human communication, navigation headings, compass bearings, and everyday geometry where 360° divisions are intuitive.

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Glossary

Degree (°)
The most common unit of angle measurement, dividing a full circle into 360 parts. The 360-degree convention originates from ancient Babylonian astronomy and has been the dominant angular unit in navigation and everyday geometry ever since.
Radian (rad)
The SI unit of angle, defined as the angle subtended at the centre of a circle by an arc equal in length to the radius. A full circle is 2π radians. Radians are the natural unit for trigonometric functions and calculus.
Gradian (grad)
Also called a gon, the gradian divides a right angle into 100 parts and a full circle into 400 parts. Introduced during the French Revolution as part of the metric system, it simplifies some surveying calculations and is still used in parts of European civil engineering.
Turn (full revolution)
A unit equal to one complete rotation, equivalent to 360°, 2π radians, or 400 gradians. The turn is increasingly favoured in computer graphics and animation because it expresses rotations without requiring knowledge of π.
Degrees-Minutes-Seconds (DMS)
A notation for angles that subdivides degrees into 60 minutes (′) and each minute into 60 seconds (″). Used in geographic coordinates, navigation charts, and astronomical positions. 1 arcsecond ≈ 1/3600 of a degree.
Arc Length
The distance along a curved line or arc. For a circle of radius r, the arc length s subtended by an angle θ in radians is s = rθ. This direct relationship between arc length and radians is why radians are the natural angular unit in mathematics.