Degrees, Radians, and Gradians: Which Angle Unit to Use and When
By Convert Calc Editorial Team · Published
Angle is the one quantity where the unit changes the answer even when nobody makes an arithmetic mistake. A spreadsheet that returns 0.85 for the sine of 45 isn’t broken — it read 45 as radians. A survey bearing of 250 that looks like a plausible compass direction is 225° if the instrument was set to gon. Three units are in common use, they overlap in range, and almost nothing labels which one it’s using.
Why a circle has 360 degrees
The 360-degree circle is inherited, not derived. It comes out of the sexagesimal (base-60) counting the Babylonians took over from the Sumerians — the same counting that still gives us 60 minutes in an hour and 60 seconds in a minute, a convention that has survived some 4,000 years. The degree keeps the pattern all the way down: it splits into 60 arcminutes, and each arcminute into 60 arcseconds.
Why 360 specifically is less settled than popular accounts suggest. The usual explanations — that the year was taken to be 360 days, or that 60 was picked because it has an unusually rich set of divisors — are reconstructions historians argue over, not documented decisions. What is certain is the practical consequence: 360 divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180. Halves, thirds, quarters, fifths, sixths, eighths, and twelfths of a circle are all whole numbers of degrees. That is why degrees survive in navigation, construction, and machining: the angles people actually use come out clean.
The degree isn’t an SI unit, but it’s formally accepted for use with SI, defined against the radian as 1° = π/180 rad (NIST SP 330).
What a radian actually is
The radian is defined by the circle itself. Take a circle, measure off an arc as long as the radius, and the angle that arc subtends at the center is one radian (NIST SP 330, Table 4 footnote). That makes the radian a ratio — arc length divided by radius, metre over metre — which is why it’s the coherent SI unit of plane angle and why it’s dimensionless.
Two consequences follow directly, and they’re the reason radians dominate technical work:
- Arc length is just rθ. A 2.4 m radius turning through 0.75 rad sweeps 1.8 m of arc. No conversion constant. In degrees the same calculation needs a π/180 factor bolted on.
- Calculus comes out clean. The derivative of sin(x) is cos(x) only when x is in radians. The whole result rests on sin(θ)/θ approaching 1 as θ approaches zero, and that limit is 1 only in radian measure — because on a unit circle a small radian angle and the sine of that angle are very nearly the same number. Work in degrees and the limit becomes π/180, and that stray factor then propagates into every derivative, every integral, and every series expansion downstream.
This is also why every standard math library — Python’s math.sin(), JavaScript’s Math.cos(),
C’s sin() — takes radians and returns radians. Those libraries implement the calculus-era
definitions, and the calculus-era definitions assume radians.
The practical cost is that radians are ugly to read. A right angle is 1.5708. A third of a circle is 2.0944. Nothing lands on a round number except zero, which is exactly the trade you make for the clean derivatives.
Gradians: the decimal angle that didn’t catch on
The gradian — standard symbol gon — divides the right angle into 100 parts, so a full circle is 400 gon. It came out of the same French revolutionary push toward decimalization that produced the metric system, and it’s the one piece of that decimalization program that never went global. Decimal time died; the gon didn’t.
It survives because it suits surveying. Bearings are quadrant work, and quadrant work in gon is decimal arithmetic: 100 gon per quadrant, no sexagesimal arcminutes and arcseconds to carry through a traverse calculation, no degrees-minutes-seconds parsing when data moves between instruments. Theodolites and total stations ship with a gon mode, and in parts of continental Europe that’s the mode they’re set to. The gon is a legal unit of measurement in the EU, listed alongside the degree in the EU units of measurement directive (under both names, “grade” and “gon”).
If you never touch European survey data or a total station, you can ignore gradians, with one exception: scientific calculators put a GRAD mode right next to DEG and RAD, and it’s the easiest of the three to land in by accident.
Reference table
All three units describe the same angles. Only the labelling differs.
| Fraction of circle | Degrees | Radians (exact) | Radians (decimal) | Gradians (gon) |
|---|---|---|---|---|
| 0 | 0° | 0 | 0 | 0 |
| 1/24 | 15° | π/12 | 0.2618 | 16.667 |
| 1/12 | 30° | π/6 | 0.5236 | 33.333 |
| 1/8 | 45° | π/4 | 0.7854 | 50 |
| 1/6 | 60° | π/3 | 1.0472 | 66.667 |
| 1/4 | 90° | π/2 | 1.5708 | 100 |
| 1/3 | 120° | 2π/3 | 2.0944 | 133.333 |
| 1/2 | 180° | π | 3.1416 | 200 |
| 3/4 | 270° | 3π/2 | 4.7124 | 300 |
| 1 | 360° | 2π | 6.2832 | 400 |
The conversions themselves are one multiplication each:
| From → to | Multiply by | Note |
|---|---|---|
| Degrees → radians | π/180 ≈ 0.0174533 | exact by definition |
| Radians → degrees | 180/π ≈ 57.295780 | exact by definition |
| Degrees → gradians | 10/9 ≈ 1.111111 | exact |
| Gradians → degrees | 0.9 | exact |
| Radians → gradians | 200/π ≈ 63.661977 | exact by definition |
Unit definitions follow ISO 80000-3:2019, the standard this site uses for all angle conversions.
The mode mismatch, which is what actually bites people
Nearly every real angle error traces back to one of a handful of places where the unit is implied rather than stated.
Calculators. DEG, RAD, and GRAD are a mode setting, and the indicator is a three-letter label in a corner of the display that’s easy to miss. Test it before you trust it: sin(90) is 1 in DEG, roughly 0.894 in RAD, and roughly 0.988 in GRAD. That last one is the dangerous case — 0.988 looks near enough to 1 to pass a glance, which is how a GRAD-mode calculator survives an entire problem set undetected.
Spreadsheets. Excel, Google Sheets, and LibreOffice have no mode. SIN(), COS(), and
TAN() always take radians, and ASIN(), ACOS(), ATAN(), and ATAN2() always return them.
Wrap the input in RADIANS() and the output in DEGREES(), every time: =SIN(RADIANS(A2)), not
=SIN(A2).
Programming languages. Same rule, no mode, no warning. math.sin(45) in Python is the sine
of 45 radians (about 0.851), not of 45 degrees (about 0.707). Both are valid numbers in range,
so nothing throws, nothing logs, and the error shows up much later as a model that’s subtly wrong.
CAD, GIS, and instrument exports. Angles cross these boundaries as bare numbers. A CAD file
in degrees, a total station in gon, and a script calling atan2() in radians can all appear in
the same workflow with no unit recorded anywhere in the data.
Two habits prevent most of this. Put the unit in the variable or column name —
heading_deg, theta_rad — so a mismatch is visible when you read the line rather than when you
read the output. And sanity-check against a known value: feed the function 90 (or π/2, or 100 gon)
and confirm you get the answer you expect before running the real numbers.
Frequently Asked Questions
My spreadsheet says the sine of 45 is 0.851, but it should be 0.707. What’s wrong?
Nothing is wrong with the spreadsheet — SIN() takes radians, and it computed the sine of 45
radians. Use =SIN(RADIANS(45)) to get 0.7071. The same applies in reverse: ASIN() hands back
radians, so wrap it in DEGREES() if you want a degree answer.
My calculator gives sin(90) = 0.988 instead of 1. Is it broken?
It’s in GRAD mode, where 90 means 90 gon — that’s 81°, and sin(81°) = 0.9877. Press MODE and switch to DEG. The near-miss is what makes this one so easy to skip past; check the mode indicator whenever a trig answer is almost right.
A European survey lists a bearing of 250. Is that degrees or gradians?
Check the instrument setting or the project’s units note before assuming. If it’s gon, 250 gon is 225°. Both values are valid bearings, so the number itself won’t tell you — but a dataset where angles run past 360 is certainly in gon, since a full circle is 400 gon.
Do I need radians if I never do calculus?
If you write code or use spreadsheets, yes — the math functions take radians whether or not you’re doing calculus. You don’t need to think in radians; you just need to convert on the way in and on the way out.
Why does my robot arm move to the wrong position when I enter the datasheet angle?
Datasheets specify joint angles in degrees; most control firmware and motion libraries (ROS included) take radians. Entering 37.5 where 37.5° was intended commands about 2,149° of rotation — which the controller will clamp, wrap, or fault on depending on the driver. Convert first: 37.5 × π/180 ≈ 0.6545 rad.
Is the radian a “real” unit if it’s dimensionless?
Yes. It’s the coherent SI unit of plane angle, defined as the ratio of arc length to radius (m/m). Being dimensionless means it cancels cleanly in equations like s = rθ — which is the property that makes it useful, not a sign that it’s somehow lesser than the degree.
Practical takeaway
- Degrees for anything a person reads: bearings, construction drawings, machining specs, weather and navigation.
- Radians for anything a machine computes: trig functions, calculus, physics, rotation matrices, every standard math library.
- Gradians only when the data source uses them — European survey records and instruments set to gon mode.
Before trusting any trig result, confirm which unit the tool expects. Check the calculator’s mode
indicator, wrap spreadsheet trig in RADIANS() and DEGREES(), and name variables with their
unit. For the conversions themselves, see
degrees to radians,
radians to degrees, and
gradians to degrees.