Angle Converter
Type an angle in any unit, even 45°30′15″ or N 45° E, and see it in degrees, radians, DMS, gradians, turns, arcminutes, arcseconds and mils at once.
Type 45, 45.5, 45°30′15″ or 45 30 15, π/4, 1.2 rad, 100 gon, a bearing like N 45° E, a compass point like SSW, or 33° 52′ S (south and west count as minus). Your units are remembered on this device.
Type a slope or a ratio (1:12 or just 12) and the angle box follows; change the angle and they follow it. The clock time gives the angle between the hour and minute hands.
Drawn maths-style: 0° points right and angles turn anticlockwise.
| Degrees | π form | Radians | Gon | Turns | NATO mils | DMS |
|---|---|---|---|---|---|---|
| 0° | 0 | 0 | 0 | 0 | 0 | 0° |
| 1° | π/180 | 0.0175 | 1.1111 | 0.0028 | 17.78 | 1° |
| 5° | π/36 | 0.0873 | 5.5556 | 0.0139 | 88.89 | 5° |
| 10° | π/18 | 0.1745 | 11.1111 | 0.0278 | 177.78 | 10° |
| 15° | π/12 | 0.2618 | 16.6667 | 0.0417 | 266.67 | 15° |
| 22.5° | π/8 | 0.3927 | 25 | 0.0625 | 400 | 22° 30′ |
| 30° | π/6 | 0.5236 | 33.3333 | 0.0833 | 533.33 | 30° |
| 45° | π/4 | 0.7854 | 50 | 0.125 | 800 | 45° |
| 57.2958° | 1 rad | 1 | 63.662 | 0.1592 | 1,018.59 | 57° 17′ 44.81″ |
| 60° | π/3 | 1.0472 | 66.6667 | 0.1667 | 1,066.67 | 60° |
| 72° | 2π/5 | 1.2566 | 80 | 0.2 | 1,280 | 72° |
| 90° | π/2 | 1.5708 | 100 | 0.25 | 1,600 | 90° |
| 120° | 2π/3 | 2.0944 | 133.3333 | 0.3333 | 2,133.33 | 120° |
| 135° | 3π/4 | 2.3562 | 150 | 0.375 | 2,400 | 135° |
| 150° | 5π/6 | 2.618 | 166.6667 | 0.4167 | 2,666.67 | 150° |
| 180° | π | 3.1416 | 200 | 0.5 | 3,200 | 180° |
| 225° | 5π/4 | 3.927 | 250 | 0.625 | 4,000 | 225° |
| 270° | 3π/2 | 4.7124 | 300 | 0.75 | 4,800 | 270° |
| 300° | 5π/3 | 5.236 | 333.3333 | 0.8333 | 5,333.33 | 300° |
| 315° | 7π/4 | 5.4978 | 350 | 0.875 | 5,600 | 315° |
| 360° | 2π | 6.2832 | 400 | 1 | 6,400 | 360° |
How your angle is converted
Your numbers, step by step. Every angle unit is a fixed slice of one full turn, so the converter goes through degrees.
- 1
Your angle
45°
in degrees - 2
Make it degrees
Already degrees
45° - 3
Times the factor
45 × π/180
= 0.785398 rad - 4
Every unit
π/4 rad · 50 gon
800 NATO mils
Read more: the formula, a worked example, mils and tables
The formula
degrees = your number × degrees in one of your unit
answer = degrees ÷ degrees in one of the unit you want
radians = degrees × π ÷ 180
degrees = radians × 180 ÷ π
gon = degrees × 10 ÷ 9
decimal degrees = D + M ÷ 60 + S ÷ 3600
NATO mils = degrees × 6400 ÷ 360
slope % = tan(angle) × 100
Worked example
45° in radians: 45 × π ÷ 180 = π ÷ 4 = 3.14159265 ÷ 4 = 0.785398 rad. The same 45° is:
- Gradians: 45 × 10 ÷ 9 = 450 ÷ 9 = 50 gon.
- Turns: 45 ÷ 360 = 0.125 turn (one eighth of a circle).
- Arcminutes: 45 × 60 = 2,700′; arcseconds: 45 × 3,600 = 162,000″.
- NATO mils: 45 × 6,400 ÷ 360 = 288,000 ÷ 360 = 800 mils; Warsaw Pact: 45 × 6,000 ÷ 360 = 750; true milliradians: 0.785398 × 1,000 = 785.398 mrad.
And from DMS: 45° 30′ 15″ is 45 + 30 ÷ 60 + 15 ÷ 3,600 = 45 + 0.5 + 0.0041667 = 45.504167°, which is 45.504167 × π ÷ 180 = 0.794198 rad.
Handy angle factors
| 1 of this | Equals | In one full turn |
|---|---|---|
| Degree | π/180 = 0.0174533 rad | 360 |
| Radian | 180/π = 57.2958° = 57° 17′ 44.81″ | 2π = 6.283185 |
| Gradian (gon) | 0.9° (exact) | 400 |
| Turn | 360° = 2π rad | 1 |
| Arcminute (′) | 1/60° = 60″ | 21,600 |
| Arcsecond (″) | 1/3600° | 1,296,000 |
| Mil (NATO) | 0.05625° (exact) | 6,400 |
| Mil (Warsaw Pact) | 0.06° | 6,000 |
| Streck (Sweden) | 0.0571429° | 6,300 |
| Milliradian (true) | 0.001 rad = 0.0572958° | 2,000π = 6,283.19 |
Which mil is which?
“Mil” means different things. NATO armies split a circle into 6,400 mils because it halves neatly again and again (6,400, 3,200, 1,600, 800 …) and is close to the true milliradian. The old Soviet and Warsaw Pact system uses 6,000, and Sweden used 6,300 streck. Rifle scopes and rangefinders marked “mil” or MRAD use the true milliradian, a thousandth of a radian, so there are about 6,283.19 in a circle. At 1,000 m, 1 true milliradian covers 1 m; a NATO mil covers about 0.98 m.
Bearings, compass points and hemispheres
A bearing like N 45° E starts from north and turns 45° towards east, so it is 45° on a 360° compass. S 45° E is 180 − 45 = 135°, S 30° W is 180 + 30 = 210°, and N 10° W is 360 − 10 = 350°. A 16-point compass steps 22.5° (NNE is 22.5°), and a 32-point compass steps 11.25°. For map coordinates, south latitudes and west longitudes count as minus: 33° 52′ S is −33.8667°.
Slope and ratio
slope % = tan(angle) × 100
angle = arctan(slope % ÷ 100)
angle = arctan(rise ÷ run)
roof pitch = 12 × tan(angle) in 12
A 1:12 ramp rises 1 for every 12 along: arctan(1 ÷ 12) = 4.7636°, a slope of 8.33%. A 100% slope is 45°, not 90°: it rises as much as it runs. A vertical wall has no slope percentage at all.
Clock hands
hour hand = 30 × hours + 0.5 × minutes
minute hand = 6 × minutes
angle = the gap, or 360 minus the gap, whichever is smaller
At 3:15 the hour hand is at 90 + 7.5 = 97.5° and the minute hand at 90°, so the angle is 7.5°. At 10:10 it is |305 − 60| = 245°, so the smaller angle is 360 − 245 = 115°.
More converters and tools: unit converter, speed converter and the trigonometry calculator for sin, cos and tan of any angle.
All factors are exact by definition (π radians = 180°, 400 gon = 360°); mil conventions as used by NATO (STANAG 6400), the former Warsaw Pact (6000) and Sweden (6300). Written 14 October 2026.
Common questions
How do I convert degrees to radians?
Multiply the degrees by π and divide by 180. So 45° is 45 × π ÷ 180 = π/4, which is 0.785398 radians, and 90° is π/2 or 1.570796 radians. To go the other way, multiply radians by 180 and divide by π: 1 radian is about 57.2958°.
How do I convert DMS to decimal degrees?
Add the minutes divided by 60 and the seconds divided by 3,600 to the degrees. So 45° 30′ 15″ is 45 + 30 ÷ 60 + 15 ÷ 3,600 = 45 + 0.5 + 0.004167 = 45.504167°. For a south latitude or west longitude, put a minus sign in front. You can type 45 30 15 or 45°30′15″ in the box above.
What is a gradian (gon)?
A gradian, also called a gon or grad, is one 400th of a full turn, so a right angle is exactly 100 gon and 1 gon is 0.9°. Surveyors in parts of Europe use it. To change degrees to gon, multiply by 10 and divide by 9: 45° is 50 gon.
How many mils are in a circle?
It depends on who is counting. NATO uses 6,400 mils in a circle, so 1 mil is 0.05625°. The old Warsaw Pact (and Russia) uses 6,000, and Sweden used 6,300. A true milliradian is one thousandth of a radian, so there are 2,000π, about 6,283.19, in a circle. Rifle scopes marked “mil” or MRAD use the true milliradian.
How do I change a slope percentage into degrees?
Divide the percentage by 100 and take the inverse tangent (arctan). A 100% slope is 45°, a 10% hill is arctan(0.1) = 5.71°, and a 1:12 wheelchair ramp (8.33%) is 4.76°. Switch to Advanced to type a slope or a ratio and see the angle.
What angle do the clock hands make at 3:15?
7.5°. The minute hand is at 15 × 6 = 90°, but the hour hand has moved a quarter of the way from 3 to 4, so it is at 3 × 30 + 15 × 0.5 = 97.5°. The gap is 97.5 − 90 = 7.5°. The clock tool in Advanced works it out for any time.
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