Angle Converter
Degrees, radians, gradians, turns, arcminutes, arcseconds and milliradians — with degrees-minutes-seconds both in and out, because that is how latitude and longitude are written. Every answer is shown decimally and sexagesimally at once.
Convert an angle
Latitude and longitude are usually written the second way. The answer is shown both ways regardless.
The angle you have, in the unit chosen below. Negative values are allowed — a bearing correction has a direction.
The unit the angle is already in.
The unit you want the answer in. Every other unit — and the sexagesimal form — is listed underneath the result anyway.
Routed through the radian: 45° is 0.7853981634 rad. Written sexagesimally, the same angle is 45° 00′ 00″.
- Degrees, minutes, seconds
- 45° 00′ 00″
- Decimal degrees
- 45°
- Conversion factor
- 1° = 0.01745329252 rad
- Reverse factor
- 1 rad = 57.29577951°
- In radians
- 0.7853981634 rad
- Definition
- The SI unit: the angle that subtends an arc equal to the radius. Arc length is r × θ, and the derivative of sin θ is cos θ, only when θ is in radians.
| Unit | Symbol | Value |
|---|---|---|
| Arcseconds | ″ | 162,000 |
| Arcminutes | ′ | 2,700 |
| Milliradians | mrad | 785.3981634 |
| Gradians (gon) | gon | 50 |
| Degrees | ° | 45 |
| Radians | rad | 0.7853981634 |
| Turns (revolutions) | turn | 0.125 |
Degrees, minutes, seconds — base 60, still in daily use
A degree splits into 60 minutes of arc, and each of those into 60 seconds of arc. The names are borrowed from time for the same reason time uses them: both descend from Babylonian base-60 arithmetic, and both survived the metric reform that decimalised nearly everything else. “Minute” is the pars minuta prima, the first small part; “second ” is the pars minuta secunda.
This is not a museum piece. Every nautical chart, aviation waypoint, land survey, telescope pointing and street address in a cadastral system is written in degrees, minutes and seconds. So the conversion is worth doing carefully rather than approximately:
- The carry is not optional. Seconds have to be rounded somewhere, and a rounded 59.99996″ becomes 60.0000″ — which is not a legal seconds value. It has to carry into the minute, and an overflowed minute into the degree. Skip that and the tool prints 41° 59′ 60″, which is 42° written wrongly.
- The sign does not live in the degrees. Half a degree south of the equator is −0° 30′ 00″, and there is no such integer as −0. This converter carries the sign as its own flag and prints it in front of the whole reading.
- Over-full entries are normalised, not refused. Type 41° 24′ 60″ and you get 41° 25′ 00″ back. That is what you meant, and watching the carry happen is the clearest possible demonstration of what base 60 actually does.
Every conversion goes through the radian
Seven units could be connected by 49 direct factors. Instead each is stored as one number — how many radians it is — and every conversion is two steps.
radians = value × f(from) • result = radians ÷ f(to)- f(u)
- Radians in one unit u — π/180 for the degree
- radians
- The angle in the SI unit
- result
- The angle in the unit you asked for
The radian is the pivot rather than the degree for the reason calculus prefers it. The radian is defined by the circle itself: it is the angle that subtends an arc equal in length to the radius. That is what makes arc length simply r × θ, and what makes the derivative of sin θ equal cos θ with no stray constant. Every other unit here is a historical convention scaled off a full turn.
- 1 turn = 2π rad = 360° = 400 gon, exactly. The turn is the only unit whose definition needs no reference to any other.
- 1° = π/180 rad, and an arcminute and arcsecond are a sixtieth and a 3,600th of that.
- 1 mrad = 0.001 rad, exactly — a decimal subdivision of the SI unit, unlike the artillery mils it is often confused with.
Decimal results are rounded to twelve significant figures rather than a fixed number of decimal places, because a table spanning arcseconds to turns has no single sensible decimal place. The DMS output is different in kind: degrees and minutes come back as exact integers, and only the seconds field carries a rounded fraction — four decimal places, about 3 mm of latitude.
45 degrees, and a latitude
Start with the default. 45 × π/180 = 0.7853981634 radians, which is π/4. The same angle is 50 gon (45 × 400/360), 0.125 turn (45/360), 2,700 arcminutes and 162,000 arcseconds. Only one of those five needed π; the rest are exact fractions of a circle, and they come back as whole numbers because they genuinely are whole numbers.
Now the sexagesimal path. Switch the input to degrees-minutes-seconds and enter 41° 24′ 12.2″ — a latitude in central Italy. Minutes first: 24 ÷ 60 = 0.4. Then seconds: 12.2 ÷ 3600 = 0.003388889. Add them to 41 and the decimal degree is 41.40338889°, which is 0.7226254576 radians. Read the DMS line underneath and it says 41° 24′ 12.2″ again — the round trip is the check.
One more, to see the carry work. Enter 41° 24′ 60″. Sixty arcseconds is a full arcminute, so the answer normalises to 41° 25′ 00″ rather than being rejected. Enter 41.99999999 in decimal degrees and the seconds field rounds to 60.0000, carries into the minute, which is then 60 and carries again: 42° 00′ 00″.
What this converter assumes
- An angle is a plain plane angle. Nothing here normalises to a range: 450° stays 450° rather than becoming 90°, because a rotation count and a heading are different things and the tool cannot tell which you meant.
- Degrees-minutes-seconds input is a magnitude. Signs go in via decimal mode, because coordinates in the wild carry their sign as N/S or E/W rather than as a minus.
- The milliradian is the true one, 0.001 rad. The NATO, Warsaw Pact and Swedish artillery mils — 6,400, 6,000 and 6,300 to the turn — are deliberately absent, because offering four things all called “mil” would create more errors than it fixes.
- Compass points are not units. Rounding an angle to the nearest of 16 or 32 named sectors destroys precision silently, so N, NNE and the rest are not offered.
- Seconds are reported to four decimal places, roughly 3 mm of latitude. That is a display choice, not a claim about your measurement — a GPS fix good to 3 m is still good to 3 m after conversion.
Angle conversion FAQ
How do I convert degrees-minutes-seconds to decimal degrees?
Divide the minutes by 60, divide the seconds by 3,600, and add both to the degrees. 41° 24′ 12.2″ becomes 41 + 24/60 + 12.2/3600 = 41.40338889°. Going back the other way, take the whole degrees, multiply what is left by 60 to get whole minutes, and multiply what is left of that by 60 to get seconds. The one step people miss is the carry: round the seconds to a few decimal places and they can land on exactly 60, which then has to become 0 with a minute added.
Why does a circle have 360 degrees?
Two reasons that reinforced each other. The Babylonians counted in base 60, and their astronomy divided the zodiac into 12 signs of 30 parts — 360 in all, conveniently close to the number of days in a year, so the sun moves roughly one degree a day. And 360 is unusually divisible: it splits evenly 24 ways, including into halves, thirds, quarters, fifths, sixths, eighths, ninths, tenths and twelfths. A hundred-part circle would have no whole-number third and no whole-number eighth, which is why the metric attempt on the circle never displaced it.
Why do mathematicians use radians instead of degrees?
Because the radian is the only unit in which the geometry comes out clean. An arc of a circle has length r × θ only when θ is in radians; in degrees you need an extra factor of π/180. The same factor turns up in calculus: the derivative of sin θ is cos θ in radians and (π/180)·cos θ in degrees, and the small-angle approximation sin θ ≈ θ only holds in radians. A degree is a human convention scaled off a full turn. A radian is what the circle itself measures with — the angle whose arc equals its radius.
What is a gradian, and does anyone use it?
A gradian, or gon, is a 400th of a turn, so a right angle is exactly 100 of them and a quadrant of the Earth is 100 gon of latitude. It came out of the same French revolutionary push that produced the metre — decimalise everything, including the circle. It failed almost everywhere: 360 divides too well and too much existing work was in degrees. It survives in continental European surveying and on the GRAD setting of most scientific calculators, which is why a trigonometry answer occasionally comes out as though the angle had been multiplied by 0.9.
How precise is one arcsecond on the ground?
About 31 metres of latitude — the Earth's meridian circumference divided by 1,296,000 arcseconds. One arcminute of latitude is one nautical mile, 1,852 m, which is where the nautical mile came from. Longitude arcseconds shrink with the cosine of latitude: 31 m at the equator, about 23 m in Rome, and nothing at all at the poles. This converter carries seconds to four decimal places, roughly 3 mm, which is finer than any map and most surveys.
Can I enter a southern latitude or a western longitude?
Yes, but in decimal mode with a minus sign. Degrees-minutes-seconds mode takes a magnitude only, because in the wild the sign is written as a hemisphere letter — 41° 24′ 12.2″ N, not +41° 24′ 12.2″ — and a degrees field of 0 has nowhere to put a minus. Enter −0.5 in decimal degrees and the sexagesimal answer comes back as −0° 30′ 00″, with the sign carried outside the degrees.
Is a milliradian the same as a mil?
Not usually, and the collision causes real errors. A milliradian is exactly a thousandth of a radian, which puts 6,283.19 of them in a turn. The artillery and optics "mils" are rounded approximations of it, chosen to divide the circle neatly: NATO uses 6,400 to the turn, the former Warsaw Pact used 6,000, and Sweden used 6,300. This tool offers the true milliradian only, because it is the one that is unambiguous.
Sources and review notes
- BIPM — The International System of Units (SI) Brochure, which defines the radian and lists the degree, the minute and the second of arc among the non-SI units accepted for use with the SI
- NIST Special Publication 811 — Guide for the Use of the International System of Units, for the degree-to-radian factor and for the style rules on closing up ° ′ ″ against the number
These relationships are definitions rather than measurements, so they do not go stale: a turn has been 2π radians for as long as the radian has existed, and a degree has been a 360th of a turn since Babylon. The only judgement call on this page is which mils to leave out, and that one is argued in the assumptions above.