Angle Converter
Enter an angle in degrees, radians, gradians, turns, arcminutes, arcseconds or degrees-minutes-seconds and see it in every other unit — radians as exact multiples of π (150° is 5π/6), gradians and turns as exact fractions, and the DMS split worked out to the second. Free, no sign-up; it runs in your browser.
Whole numbers, decimals and fractions such as 1/8. Negative angles and angles over a full turn are fine.
Results
| Unit | Exact | Decimal |
|---|---|---|
| Degrees | 30° | 30° |
| Radians | π/6 rad | 0.5236 rad |
| Gradians | 100/3 gon | 33.3333 gon |
| Turns | 1/12 turn | 0.0833 turn |
| Arcminutes | 1800′ | 1,800′ |
| Arcseconds | 108000″ | 108,000″ |
| Degrees, minutes, seconds | 30° 0′ 0″ | 30° 0′ 0″ |
| Coterminal angle (0° to 360°) | 30° | 30° |
The working
rad = deg × π / 180 = 30 × π / 180 = π/6
gon = deg × 10 / 9 = 30 × 10 / 9 = 100/3
turn = deg / 360 = 30 / 360 = 1/12
arcmin = deg × 60 = 1800
arcsec = deg × 3600 = 108000
Figure
How to Use the Angle Converter
- 1. Choose the unit you have. Degrees, radians, gradians (gon), turns, arcminutes, arcseconds, or degrees, minutes and seconds in three boxes.
- 2. Type the angle. Whole numbers, decimals and fractions such as 1/8 are read exactly. In radians, a multiple of π — pi/6, 5π/12 — stays exact, while a plain number such as 1 is read as radians and converted as a decimal.
- 3. Read the table. Every unit appears as an exact value where one exists (a third of a gradian, 1/12 of a turn) and as a decimal rounded to the places you choose. The angle is also reduced to the range 0° to 360°.
- 4. Check the working. The conversion factors are written out with your number, and the figure marks the quarter turns in degrees, radians and gradians beside your angle.
The Units and How They Relate
| Unit | Full turn | Right angle | Definition |
|---|---|---|---|
| Degree (°) | 360° | 90° | 1/360 of a turn |
| Radian (rad) | 2π rad | π/2 rad | The angle whose arc equals the radius |
| Gradian (gon) | 400 gon | 100 gon | 1/400 of a turn |
| Turn | 1 | 1/4 | One full revolution |
| Arcminute (′) | 21,600′ | 5,400′ | 1/60 of a degree |
| Arcsecond (″) | 1,296,000″ | 324,000″ | 1/60 of an arcminute |
| To get | Formula |
|---|---|
| Radians | degrees × π / 180 |
| Degrees from radians | radians × 180 / π |
| Gradians | degrees × 10 / 9 |
| Turns | degrees / 360 |
| Arcminutes | degrees × 60 |
| Arcseconds | degrees × 3600 |
| Decimal degrees from DMS | d + m / 60 + s / 3600 |
Common Angles
| Degrees | Radians | Gradians | Turns |
|---|---|---|---|
| 30° | π/6 | 33⅓ | 1/12 |
| 45° | π/4 | 50 | 1/8 |
| 60° | π/3 | 66⅔ | 1/6 |
| 90° | π/2 | 100 | 1/4 |
| 120° | 2π/3 | 133⅓ | 1/3 |
| 180° | π | 200 | 1/2 |
| 270° | 3π/2 | 300 | 3/4 |
| 360° | 2π | 400 | 1 |
Worked Examples
- Degrees to radians. 150° × π / 180 = 5π/6 rad ≈ 2.6180.
- Radians to degrees. 3π/4 rad × 180 / π = 135°.
- One radian. 1 rad = 180 / π ≈ 57.2958°, which is 57° 17′ 44.81″.
- Decimal degrees to DMS. 40.7128° has 0.7128 × 60 = 42.768′, and 0.768′ × 60 = 46.08″: 40° 42′ 46.08″.
- DMS to decimal degrees. 40° 26′ 46″ = 40 + 26/60 + 46/3600 ≈ 40.4461°, or 145,606 arcseconds.
- Gradians. A right angle is 100 gon, so 50 gon is 45° and 1 gon is 0.9°.
Where Each Unit Is Used
Degrees are everyday units for compass bearings, geometry and map latitudes. Radians are the natural unit of calculus and physics: with θ in radians, the arc length is rθ and the derivative of sin θ is cos θ. Gradians (also called gon) are used in surveying and some European engineering. Turns appear in CSS animations and rotation parameters in code. Arcminutes and arcseconds measure small angles in astronomy, optics and navigation, and DMS is the traditional way to write latitudes and longitudes.
Frequently Asked Questions
- How do you convert degrees to radians?
- Multiply by π/180, because 180° equals π radians. So 90° is π/2, 60° is π/3 and 150° is 5π/6. Keep the π in the answer for an exact value, or multiply it out (π ≈ 3.14159) for a decimal.
- How do you convert radians to degrees?
- Multiply by 180/π. π/6 radians is 30° and 2π radians is 360°. A plain number of radians such as 1 gives a decimal: 1 rad ≈ 57.2958°.
- What is a gradian?
- A gradian, or gon, is 1/400 of a full turn, so a right angle is exactly 100 gradians. It was designed to make right angles and percent gradients easy to compute and is still used in surveying in parts of Europe. Degrees convert to gradians by multiplying by 10/9.
- How do I convert decimal degrees to degrees, minutes and seconds?
- The whole number is the degrees. Multiply the decimal part by 60 for the minutes; the whole number of that is the minutes. Multiply the remaining decimal part by 60 for the seconds. For 40.7128°: 0.7128 × 60 = 42.768, so 42′, and 0.768 × 60 = 46.08″.
- How many arcseconds are in a degree?
- 3,600: there are 60 arcminutes in a degree and 60 arcseconds in an arcminute. A full circle is therefore 1,296,000 arcseconds.
- Why does 1 radian equal about 57.2958 degrees?
- A radian is the angle at the center of a circle that cuts off an arc as long as the radius. The whole circumference is 2π radii, so a full turn is 2π radians, and 360° / 2π = 180° / π ≈ 57.2958° per radian.
- What does the coterminal angle mean?
- Angles that differ by whole turns end at the same place: 750°, 390° and 30° are coterminal. The converter shows the equivalent angle from 0° up to (but not including) 360°, found by adding or subtracting full turns.