Convert angles between decimal degrees, degrees-minutes-seconds (DMS), radians, gradians, turns, arcminutes, arcseconds and milliradians.
By AdminDrafted with AI writing toolsUpdated Checked examples: 8
12°30'15" is 12.50416667° (12° 30′ 15″), which is 0.21823888 rad, 13.89351852 grad or 0.0347338 of a turn.
Degrees + minutes/60 + seconds/3600.
12° 30′ 15″ (seconds rounded to at most 4 decimals).
Type an angle in any of seven units and it is converted to all the others. Degrees accept decimals or degrees, minutes and seconds: 1° = 60′ = 3,600″, so 12°30′15″ is 12 + 30/60 + 15/3600 = 12.5041667°. Radians use 180° = π rad, gradians 400 to a full turn and milliradians 1,000 to a radian. When the angle is a simple fraction of π, the π form is shown too, such as 5π/72 for 12.5°.
The default, 12°30′15″, is 0.2182389 rad or 45,015″. GPS coordinates and survey bearings are the usual case: a latitude written 33°51′35″ S becomes −33.8597222° for a spreadsheet or mapping tool, with south and west negative.
Only the last DMS part may carry decimals, and minutes and seconds must be below 60. Seconds are rounded to four decimal places, about 3 mm of latitude on the ground.
Divide the minutes by 60 and the seconds by 3,600, then add them to the degrees: decimal degrees = D + M/60 + S/3600. So 25°15′36″ = 25 + 0.25 + 0.01 = 25.26°. The rule is the same for latitudes, longitudes, bearings and any other angle.
Reverse the steps, carrying each leftover fraction down a level:
The result is 25°15′36″.
Multiply degrees by π/180 to get radians, and radians by 180/π to get degrees. So 25.26° ≈ 0.440870 rad, and 1 rad ≈ 57.2957795°, or 57°17′44.8″.
One full turn is 360°, 2π radians or 400 gradians. A degree is 60 arcminutes (′) or 3,600 arcseconds (″), and a milliradian is a thousandth of a radian, about 0.0573°. The NIST angle-unit table gives these relations.
| Input | Selected unit | Decimal degrees | Another representation |
|---|---|---|---|
| 12°30'15" | Degrees (decimal or DMS) | 12.5041666667 | 12° 30′ 15″ |
| 12.5 | Degrees (decimal or DMS) | 12.5 | 5π/72 rad |
| pi/4 | Radians | 45 | π/4 rad |
| 0.25 | Turns | 90 | π/2 rad |
| 400 | Gradians (gon) | 360 | 1 turn |
| -0°30' | Degrees (decimal or DMS) | −0.5 | −0° 30′ 0″ |
Each input is read in the unit beside it. 0.25 of a turn is 90°, but 0.25 as degrees is a quarter of one degree, 15′.
A minus sign applies to the whole angle: −0°30′ is −(0 + 30/60) = −0.5°. In decimal coordinates, south latitudes and west longitudes are negative, so 33°51′35″ S becomes −33.8597222°.
An uppercase N, S, E or W can go before or after the angle; N and E count as positive, S and W as negative. Use the letter or a minus sign, not both: −33°51′35″ S cancels out to +33.8597222°.
Two angles can point the same way without being equal. −0.5° faces the same direction as 359.5°, and 720°, two full turns, faces the same way as 0°; keep the full value when counting rotations.
Two angles are complementary when they add to 90° and supplementary when they add to 180°. 30° and 150° are supplementary; in radians they are π/6 and 5π/6. OpenStax's angle lesson explains complementary and supplementary angles in more detail.
12°30'15", 12 30 15, 12:30:15 and 12d 30m 15s all mean 12.5041666667°.12°30.5' works and 12.5°30' does not.pi/4 are accepted.For sine, cosine and tangent of an angle, use the trig functions calculator. For the angles of a regular polygon, use the polygon angle calculator.
Checked against: 12 + 30/60 + 15/3600 = 3001/240; ×10/9 for gradians
Checked against: degrees(1) = 57.29577951308232; 0.2957795…×60 = 17.7467707…′, 0.7467707…×60 = 44.80625″; 1/(2π)
Checked against: π rad = 180° = 200 grad by definition
Checked against: 400 grad = 1 turn = 360° by definition
Add the minutes divided by 60 and the seconds divided by 3,600 to the whole degrees: 12°30′15″ = 12 + 0.5 + 0.0041667 = 12.5041667°. For a south latitude or a west longitude, make the result negative, so 33°51′35″ S is −33.8597222°. The sign applies to the whole angle, not only to the degrees.
Keep the whole number as degrees, multiply the fractional part by 60 for minutes, then multiply the new fractional part by 60 for seconds. For 57.2957795°: 0.2957795 × 60 = 17.7467707′, and 0.7467707 × 60 = 44.806″, giving 57°17′44.806″. That angle is one radian.
One radian is 180/π ≈ 57.2958°, or 57°17′44.8″. It is the angle at the centre of a circle cut off by an arc as long as the radius, so a full turn is 2π ≈ 6.2832 rad. The radian is the SI unit of plane angle; the degree, minute and second are accepted for use with the SI (BIPM SI Brochure, Table 8).
A gradian, or gon, is 1/400 of a full turn, so a right angle is exactly 100 grad and 1 grad = 0.9°. It is used mainly in surveying in parts of continental Europe, where splitting the right angle into 100 parts suits decimal arithmetic. 200 grad equals 180°, or π rad.
A milliradian (mrad) is one thousandth of a radian, about 0.0573° or 3.44 arcminutes. At a distance of 1,000 m it spans about 1 m, which is why rifle scopes use it for holdover and range estimates. The NATO mil used for artillery is a rounded version: 6,400 to a circle, against 6,283.2 true milliradians.
This calculator includes 8 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.