Angle Converter | Degrees, Radians, Gradians & More
Convert angles between degrees, radians, gradians, turns, arcminutes, arcseconds, and milliradians. Includes DMS mode and a visual angle display.
Common Angles
What Is the Angle Converter | Degrees, Radians, Gradians & More?
This angle converter handles seven angle units simultaneously and includes a degree-minute-second (DMS) input mode, the format used in GPS coordinates, navigation, and astronomy. Enter a value once and all conversions update instantly.
What this converter includes:
- ▸7 angle units: Degrees, Radians, Gradians, Turns, Arcminutes, Arcseconds, and Milliradians
- ▸DMS input mode: Enter degrees, minutes, and seconds separately, perfect for GPS and surveying work
- ▸Common angle presets: One-click shortcuts for 0°, 30°, 45°, 60°, 90°, 180°, 270°, and 360°
- ▸Visual angle display: A real-time SVG diagram showing the angle on a circle
- ▸Copy buttons: Copy any individual result to clipboard with one click
Formula
All angle units convert through degrees as a common base. Multiply or divide by the factors below:
rad = deg × π / 180
deg = rad × 180 / π
π ≈ 3.14159265
grad = deg × 10 / 9
deg = grad × 9 / 10
Right angle = 90° = 100 grad
arcmin = deg × 60
arcsec = deg × 3600
1° = 60' = 3600"
turns = deg / 360
mrad = deg × π / 0.18
1 turn = 360° = 2π rad
How to Use
- 1
Choose your input mode
Select "Decimal Input" to enter a single number (e.g., 1.5 radians), or switch to "Degrees° Minutes' Seconds"" mode for GPS-style coordinates like 40° 26' 46".
- 2
Enter the angle value
Type the numeric value in the input field. If using DMS mode, fill in degrees, arcminutes, and arcseconds in separate boxes.
- 3
Select the unit (decimal mode only)
Use the "From Unit" dropdown to specify which unit your value is in, degrees, radians, gradians, turns, arcminutes, arcseconds, or milliradians.
- 4
Use common angle presets
Click any preset button (30°, 45°, 60°, 90°, etc.) to instantly load a common angle and see all conversions without typing.
- 5
Read results and copy
All converted values appear below along with the DMS representation and a visual diagram. Click "Copy" next to any value to copy it to your clipboard.
Example Calculation
Example 1 | Converting 90°
Example 2 | Converting 1 Radian
Example 3 | GPS Coordinate 40° 26' 46"
Understanding Angle Converter | Degrees, Radians, Gradians & More
The Seven Angle Units, What Each One Measures
Angles can be expressed in several units, each suited to a different field of work. Understanding what each unit means helps you pick the right one and avoid errors when moving between disciplines.
- ▸Degrees (°): The most familiar unit. A full circle is 360°. Used in everyday geometry, navigation, and most school-level mathematics.
- ▸Radians (rad): The SI unit of angle. One radian is the angle where the arc length equals the radius. A full circle is 2π rad ≈ 6.2832 rad. Standard in calculus, physics, and engineering formulas.
- ▸Gradians (grad): A metric-style angle unit where a full circle is 400 grad. A right angle is exactly 100 grad. Used in some European surveying and on scientific calculators.
- ▸Turns (rev): Expresses angle as a fraction of a full rotation. One turn = 360° = 2π rad = 400 grad. Useful for counting rotations in mechanical and electrical engineering.
- ▸Arcminutes ('): One sixtieth of a degree. Used in navigation, GPS coordinates, and telescopic astronomy for fine angular measurements.
- ▸Arcseconds (\"): One sixtieth of an arcminute, or 1/3600 of a degree. Used in astronomy (stellar parallax, angular diameter of objects) and precision surveying.
- ▸Milliradians (mrad): One thousandth of a radian. Used in military ballistics and rifle optics because 1 mrad ≈ 1 meter at 1,000 meters range, making distance-correction arithmetic straightforward.
History of Angle Measurement
The 360-degree circle traces back to Babylonian astronomy roughly 3,000 years ago. The Babylonians used a base-60 number system and observed that the sun moves roughly 1° per day through the sky, with approximately 360 days in a year. Their choice of 360 persisted because of its exceptional divisibility.
Radians emerged as a formal unit in the 19th century, though the concept of arc-to-radius ratio was understood earlier. The gradian was introduced during the French Revolution as part of the metric reform to give the right angle a decimal-friendly value of 100 grad.
Degrees vs. Radians, Choosing the Right Unit
The choice between degrees and radians is not arbitrary, it depends on the context:
| Context | Preferred Unit | Reason |
|---|---|---|
| Everyday geometry | Degrees | Intuitive, widely taught, easy to visualise |
| Calculus & analysis | Radians | Derivatives of trig functions work cleanly |
| Physics formulas | Radians | Angular velocity, torque, and wave equations use rad |
| Programming (Math.sin etc.) | Radians | All standard libraries expect radians |
| GPS & navigation | Degrees (DMS) | Latitude/longitude uses degrees and arcminutes |
| Surveying | Degrees or gradians | Depends on regional tradition |
| Astronomy | Degrees, arcminutes, arcseconds | Fine resolution needed for celestial positions |
| Military optics | Milliradians | 1 mrad ≈ 1 m per 1,000 m, simplifies ranging |
Common Angle Reference Table
These are the angles you will encounter most often in trigonometry, geometry, and physics:
| Degrees | Radians (exact) | Radians (decimal) | Gradians | Turns |
|---|---|---|---|---|
| 0° | 0 | 0 | 0 grad | 0 rev |
| 30° | π/6 | 0.5236 | 33.33 grad | 0.0833 rev |
| 45° | π/4 | 0.7854 | 50 grad | 0.125 rev |
| 60° | π/3 | 1.0472 | 66.67 grad | 0.1667 rev |
| 90° | π/2 | 1.5708 | 100 grad | 0.25 rev |
| 120° | 2π/3 | 2.0944 | 133.33 grad | 0.333 rev |
| 180° | π | 3.1416 | 200 grad | 0.5 rev |
| 270° | 3π/2 | 4.7124 | 300 grad | 0.75 rev |
| 360° | 2π | 6.2832 | 400 grad | 1 rev |
DMS Format, Degrees, Minutes, Seconds
DMS is the standard format for geographic coordinates. A location written as 51° 30' 26" N means 51 degrees, 30 arcminutes, and 26 arcseconds north of the equator. To convert DMS to decimal degrees:
Example: 40° 26' 46" = 40 + (26 ÷ 60) + (46 ÷ 3600) = 40 + 0.4333 + 0.01278 ≈ 40.4461°
To go from decimal degrees back to DMS: the whole number is the degrees, multiply the decimal part by 60 to get minutes, then multiply the remaining decimal by 60 again to get seconds.
Arcminutes and Arcseconds in Real-World Applications
- ▸Navigation: One arcminute of latitude = 1 nautical mile (1.852 km). This is where the nautical mile gets its definition.
- ▸Astronomy: The full moon is about 0.5° (30 arcminutes) in apparent diameter. Stars are measured in arcseconds, the nearest star beyond the Sun has a parallax of about 0.755 arcseconds.
- ▸Surveying: Total stations measure horizontal and vertical angles to arcsecond precision. A 1-arcsecond error at 1 km translates to about 5 mm of horizontal position error.
- ▸Telescopes: Resolution of optical instruments is described in arcseconds. The Hubble Space Telescope has a resolution of about 0.05 arcseconds.
Milliradians in Military and Long-Range Shooting
A milliradian (mrad or mil) is 1/1000 of a radian. Because one radian subtends an arc equal to the radius, 1 mrad subtends 1 mm at 1 metre (or 1 metre at 1,000 metres). This constant relationship makes range estimation and scope adjustment arithmetic simple: to move the point of impact 10 cm at 500 m, adjust 0.2 mrad. NATO riflescopes use mrad click values (typically 0.1 mrad per click).
Practical Uses of Angle Conversion
- ▸Converting GPS coordinates from DMS to decimal degrees for use in mapping software or databases
- ▸Converting degrees to radians before passing angles to JavaScript Math functions (Math.sin, Math.cos)
- ▸Translating physics problems between degree-based textbook answers and radian-based formula inputs
- ▸Reading surveying instrument outputs in gradians and converting to degrees for final reports
- ▸Adjusting long-range rifle scope turrets using milliradian click values
- ▸Calculating arc length and sector area from central angles expressed in different units
Frequently Asked Questions
Why do mathematicians prefer radians over degrees?
Radians make calculus formulas clean and correct. The derivative of sin(x) is cos(x) only when x is in radians. In degrees, you'd need an extra factor of π/180 every time. Radians also define arc length directly: arc length = radius × angle (in radians), with no conversion constant needed.
What are gradians, and who uses them?
A gradian (also called gon or grad) divides a full circle into 400 parts, so a right angle is exactly 100 grad. Introduced during the French Revolution as part of the metric system, gradians are still used in some European surveying traditions and appear on scientific calculators labeled "GRAD" or "G" mode.
What is a milliradian and where is it used?
A milliradian (mrad) is 1/1000 of a radian. At 1,000 meters distance, 1 mrad spans almost exactly 1 meter, making it very useful for long-range targeting and ballistics. Military rifle scopes and artillery systems often express adjustments in milliradians (mils) because the math simplifies over long distances.
What's the difference between arcminutes, arcseconds, and geographic minutes?
Angular arcminutes and arcseconds divide degrees: 1° = 60 arcminutes (') = 3600 arcseconds ('). Geographic latitude/longitude also uses this same system, one arcminute of latitude equals approximately 1 nautical mile (1.852 km). Astronomers use arcseconds to measure the apparent size of celestial objects; 1 arcsecond of parallax defines a parsec.
What is DMS format and when do I need it?
DMS stands for Degrees, Minutes, Seconds. A GPS coordinate like 40°26'46"N is in DMS format. The decimal equivalent is 40 + 26/60 + 46/3600 ≈ 40.4461°. Many mapping systems, surveying tools, and nautical charts use DMS, so being able to convert between DMS and decimal degrees is a common practical need.
How do I convert degrees to radians by hand?
Multiply the degree value by π/180. For example: 180° × π/180 = π rad. For 90°: 90 × π/180 = π/2 ≈ 1.5708 rad. To go the other way, multiply radians by 180/π.
Can I enter negative angles?
Yes. Negative angles represent clockwise rotation in standard math convention. For example, −90° is equivalent to 270° in a 0–360° range. In DMS mode, enter a negative value in the degrees field to represent a southerly latitude or westerly longitude.
Why is a full circle 360° and not 100° or 400°?
360 was chosen by ancient Babylonian astronomers, who used a base-60 (sexagesimal) number system. 360 is also highly divisible, it has 24 divisors, making it easy to split a circle into halves, thirds, quarters, sixths, eighths, and twelfths without fractions. The Babylonian calendar also had approximately 360 days per year, reinforcing the choice.