What this calculator does
Turn an angle from one unit into another, or switch map coordinates between two common formats. Angles can be measured in several ways. Degrees split a full turn into 360 parts and are the everyday choice. Radians measure an angle by the length of arc it cuts on a circle with a radius of one. That's why they show up in calculus and in programming languages. Gradians split a right angle into 100 parts and appear on surveying instruments. Arcminutes and arcseconds slice a degree into 60 and 3,600 pieces. They're used in astronomy, optics and map coordinates.
This converter has three modes. The first converts between any two angle units. The second turns a degrees-minutes-seconds (DMS) reading, such as a latitude, into decimal degrees. The third goes the other way. Every result comes with the working.
Where it's used
Why this math matters outside the classroom:
- Programming. Developers convert 45° to π/4 ≈ 0.7854 radians because most math libraries' trig functions expect radians.
- GPS coordinates. Map users turn 39° 6′ 0″ into decimal degrees as 39 + 6 ÷ 60 = 39.1°.
- Surveying. Some survey instruments read in gradians, where a right angle is 100 gon, so 90° converts to 100 gon.
- Astronomy. Astronomers describe the Moon's apparent width of about half a degree as roughly 30 arcminutes.
- Rotation counts. Engineers describe a motor shaft turning through 720° as exactly 2 turns.
Grade level and standards: Grades 10–11; Common Core State Standards for Mathematics HSG.C.B.5, HSF.TF.A.1. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Choose a mode: between angle units, DMS to decimal degrees, or decimal degrees to DMS.
- For unit conversion, enter the angle and pick the two units. Negative angles are allowed.
- For DMS to decimal, enter whole degrees, whole minutes (0–59) and seconds (less than 60). Tick the box for south latitudes or west longitudes.
- For decimal to DMS, enter the decimal degrees. Use a minus sign for south or west.
What the inputs mean
- Conversion
- Pick the kind of conversion you need.
- Angle
- The angle in the From unit. Negative values mean the angle is measured the other way (usually clockwise).
- Degrees
- Whole degrees of the DMS angle.
- Minutes
- Whole arcminutes, 0 to 59. There are 60 arcminutes in a degree.
- Seconds
- Arcseconds, at least 0 and less than 60. Decimals are allowed.
- Negative angle (south latitude or west longitude)
- Coordinates south of the equator or west of Greenwich are negative in decimal form.
- Decimal degrees
- An angle or coordinate in decimal degrees, for example −73.985656.
Formula and method
result = value × (from-unit in °) ÷ (to-unit in °) 1 rad = 180/π ° ≈ 57.29577951°; 1 gon = 0.9°; 1 turn = 360° 1′ (arcminute) = 1/60°; 1″ (arcsecond) = 1/3,600° decimal degrees = d + m/60 + s/3,600 (sign applied to the whole angle) m = whole part of (fraction of degree × 60); s = (fraction of minute) × 60
In these lines, d is the whole degrees, m is the minutes and s is the seconds. The symbol ′ means arcminutes and ″ means arcseconds. A gon is another name for a gradian, and π (pi) is about 3.1416.
A full turn is 360 degrees, 2π radians or 400 gradians. So each unit has a fixed size in degrees. Every conversion goes through degrees. That keeps degrees, arcminutes and arcseconds exact; only radians involve π.
For DMS, minutes are 60ths of a degree and seconds are 3,600ths. Going the other way, the whole number of degrees is kept. The leftover fraction is multiplied by 60 to give minutes. Then the leftover fraction of a minute is multiplied by 60 again for seconds. Seconds are rounded to four decimal places, and a rounded 60″ is carried over into the minutes.
Worked example
A map gives a longitude of −73.985656°. Write it in degrees, minutes and seconds.
- Whole degrees: 73.985656° = 73° + 0.985656°.
- Minutes: 0.985656 × 60 = 59.13936′, so 59 whole minutes.
- Seconds: 0.13936 × 60 = 8.3616″.
- Keep the negative sign for west.
Result: −73.985656° is −73° 59′ 8.3616″, or 73° 59′ 8.3616″ W.
Understanding the result
Handy landmarks: 180° is π radians (about 3.1416). A right angle is 90°, π/2 rad or 100 gon. One radian is about 57.3°.
On the Earth's surface, one arcsecond of latitude is roughly 31 meters. So four decimal places of arcseconds is far finer than any handheld GPS.
Assumptions and limitations
- A gradian is 1/400 of a turn, and a turn is one full revolution.
- DMS degrees and minutes are whole numbers. The sign applies to the whole angle.
- Angles aren't wrapped back into one turn, so 450° stays 450° rather than becoming 90°.
- Military mils aren't included. Their definition varies between 6,000 and 6,400 per turn.
- DMS output rounds seconds to four decimal places.
Frequently asked questions
How do I convert degrees to radians by hand?
Multiply by π and divide by 180. For example, 60° × π/180 = π/3, which is about 1.0472 radians. To go back, multiply radians by 180/π.
What does 40° 26′ 46″ mean?
It's 40 degrees, 26 arcminutes and 46 arcseconds. There are 60 minutes in a degree and 60 seconds in a minute. So it equals 40 + 26/60 + 46/3,600, about 40.4461 decimal degrees.
How do I enter a south or west coordinate?
In DMS mode, tick the negative box, which applies the minus sign to the whole angle. In decimal mode, type a minus sign. Putting the sign only on the degrees would give the wrong answer for something like −0° 30′.
Where are gradians used?
Mainly in surveying and on some calculators. A right angle is exactly 100 gradians, so slopes and bearings work out neatly as decimals.
How this calculator is checked
Method: Definitions of the degree (1/360 turn), radian (180/π degrees), gradian (1/400 turn) and the sexagesimal arcminute and arcsecond. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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