Angle Converter (Degrees, Radians, Gradians & DMS)

How to use the angle converter

Type an angle into any field — degrees, radians, gradians, turns, arcminutes, arcseconds, milliradians or mils — and all the others update. The last three fields split the angle into degrees, minutes and seconds (DMS), the format used for latitude and longitude. For a negative angle, convert its absolute value to DMS and put the minus sign in front.

Angle conversion formulas

rad = deg × π/180  ·  grad = deg × 10/9  ·  turns = deg/360  ·  ′ = deg × 60  ·  ″ = deg × 3600

DMS: D = ⌊deg⌋  ·  M = ⌊(deg − D) × 60⌋  ·  S = ((deg − D) × 60 − M) × 60

Worked example

Convert a latitude of 52.3725°:

• Radians = 52.3725 × π/180 = 0.914073 rad (0.290958 π rad)
• Gradians = 52.3725 × 10/9 = 58.1917 grad
• Milliradians = 914.0726 mrad
• DMS: 0.3725 × 60 = 22.35′ → 22′; 0.35 × 60 = 21″ → 52° 22′ 21″

Common angles

Degrees	Radians (π)	Radians	Gradians	Turns	NATO mils
0°	0	0.000000	0	0	0
30°	π/6	0.523599	33.3333	0.0833333	533.333
45°	π/4	0.785398	50	0.125	800
60°	π/3	1.047198	66.6667	0.166667	1066.67
90°	π/2	1.570796	100	0.25	1600
120°	2π/3	2.094395	133.333	0.333333	2133.33
135°	3π/4	2.356194	150	0.375	2400
180°	π	3.141593	200	0.5	3200
270°	3π/2	4.712389	300	0.75	4800
360°	2π	6.283185	400	1	6400

Practical notes

• Spreadsheet and programming trig functions (SIN, COS, Math.sin) expect radians — convert with RADIANS() or × π/180 first.
• One arcsecond of latitude is about 30.9 m on the ground; 0.0001° is about 11 m.
• Small-angle rule: for angles under about 10°, sin θ ≈ tan θ ≈ θ in radians — so 1 mrad ≈ 1 m at 1 km.

Degrees to radians, gradians, mils and turns

Degrees	Radians	Gradians	Mils	Turns
0°	0 rad	0 gon	0 NATO mil	0 turn
1°	0.0175 rad	1.11 gon	17.8 NATO mil	0.0028 turn
5°	0.0873 rad	5.56 gon	88.9 NATO mil	0.0139 turn
10°	0.1745 rad	11.11 gon	177.8 NATO mil	0.0278 turn
15°	0.2618 rad	16.67 gon	266.7 NATO mil	0.0417 turn
30°	0.5236 rad	33.33 gon	533.3 NATO mil	0.0833 turn
45°	0.7854 rad	50 gon	800 NATO mil	0.125 turn
60°	1.0472 rad	66.67 gon	1,066.7 NATO mil	0.1667 turn
90°	1.5708 rad	100 gon	1,600 NATO mil	0.25 turn
120°	2.0944 rad	133.33 gon	2,133.3 NATO mil	0.3333 turn
135°	2.3562 rad	150 gon	2,400 NATO mil	0.375 turn
150°	2.618 rad	166.67 gon	2,666.7 NATO mil	0.4167 turn
180°	3.1416 rad	200 gon	3,200 NATO mil	0.5 turn
210°	3.6652 rad	233.33 gon	3,733.3 NATO mil	0.5833 turn
225°	3.927 rad	250 gon	4,000 NATO mil	0.625 turn
270°	4.7124 rad	300 gon	4,800 NATO mil	0.75 turn
300°	5.236 rad	333.33 gon	5,333.3 NATO mil	0.8333 turn
315°	5.4978 rad	350 gon	5,600 NATO mil	0.875 turn
360°	6.2832 rad	400 gon	6,400 NATO mil	1 turn

Decimal degrees to degrees, minutes and seconds

Decimal degrees	DMS
12.5°	12° 30′ 0.0″
18.75°	18° 45′ 0.0″
33.333°	33° 19′ 58.8″
45.5°	45° 30′ 0.0″
51.4778°	51° 28′ 40.1″
73.9857°	73° 59′ 8.5″
90.125°	90° 7′ 30.0″

To convert, keep the whole degrees, multiply the decimal part by 60 for minutes, then multiply the remaining decimal by 60 for seconds.

Frequently asked questions

How do I convert degrees to radians?

Multiply by π/180. For example 90° × π/180 = π/2 ≈ 1.5708 rad. To convert radians to degrees, multiply by 180/π ≈ 57.2958.

How do I convert decimal degrees to degrees, minutes and seconds?

The whole number is the degrees. Multiply the decimal part by 60 — the whole number is the minutes. Multiply the remaining decimal by 60 again for the seconds. 52.3725° = 52° 22′ 21″.

What is a gradian?

A gradian (gon) is 1/400 of a turn, so a right angle is exactly 100 grad. It is used in surveying in some European countries. Check that a calculator is not in GRAD mode by mistake.

What is the difference between a milliradian and a mil?

A milliradian is exactly 1/1000 rad, so a full turn is about 6283.2 mrad. The NATO mil rounds this to 6400 per turn so that it divides evenly. Both are used for sights and rangefinding, where 1 mrad subtends about 1 m at 1 km.

How many degrees is 1 radian?

1 radian is 57.2958°, because 180° equals π radians.

How do I convert 45.5° to degrees, minutes and seconds?

45.5° is 45° 30′ 0″. The decimal 0.5 × 60 gives 30 minutes.

How many mils are in a circle?

NATO uses 6,400 mils in a full circle, so 1° is about 17.78 mils. Some other armies use 6,000 or 6,300.

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