Angles come in three units that refuse to retire: degrees for humans, radians for mathematics, and gradians for a surveying tradition that still ships on every scientific calculator. Most real confusion is not about converting between them but about not noticing which one a calculator or programming language is currently speaking, which is how sin(30) comes out as −0.988 and an afternoon disappears. This guide covers the three systems, the conversions, and that exact trap. Our free angle converter moves between all of them instantly.
In this guide
The three systems and why each exists
- Degrees split the circle into 360, a number the Babylonians liked because it divides cleanly by 2, 3, 4, 5, 6, 8, 9, 10, and 12. Human-friendly, map-friendly, protractor-friendly.
- Radians define the angle by the arc it cuts: one radian is the angle whose arc equals the radius, which makes a full circle 2π radians. The definition sounds abstract until calculus arrives, where it makes every formula simpler; radians are the native unit of mathematics, physics, and every programming language.
- Gradians split the circle into 400 so a right angle is a clean 100, a French Revolution-era attempt to decimalize angles. Surveying kept it (slopes and corners in tidy decimals), and calculators still carry the GRAD mode mostly so it can be selected by accident.
Converting between them
The bridge is the half circle: 180° = π rad = 200 grad. From it:
- degrees → radians: multiply by π/180 (≈ 0.01745)
- radians → degrees: multiply by 180/π (≈ 57.296; the mental “×57.3” is accurate to 0.007%)
- degrees → gradians: multiply by 10/9, so 1 grad = exactly 0.9°
The ×57.3 figure deserves its fame: “one radian is about 57 degrees” anchors the whole system, and remembering that a right angle is about 1.57 rad (π/2) catches most order-of-magnitude mistakes before they happen.
The table of angles you already know
| Degrees | Radians | Gradians | Where you meet it |
|---|---|---|---|
| 30 | π/6 ≈ 0.524 | 33.3 | the sin = 0.5 angle |
| 45 | π/4 ≈ 0.785 | 50 | the diagonal; slope of 100% |
| 60 | π/3 ≈ 1.047 | 66.7 | equilateral triangle corner |
| 90 | π/2 ≈ 1.571 | 100 | right angle |
| 180 | π ≈ 3.142 | 200 | straight line |
| 360 | 2π ≈ 6.283 | 400 | full turn |
Mathematicians keep radians as fractions of π rather than decimals for the same reason carpenters say “quarter inch” rather than 0.25: the fraction carries the structure. Seeing π/6 tells you instantly it is a sixth of a half-turn; 0.5236 tells you nothing.
The calculator trap
The most expensive angle mistake in the world is a mode setting. Scientific calculators have DEG, RAD, and GRAD modes; programming languages are permanently in radians. The symptom is always the same: a trig result that is confidently, weirdly wrong. The diagnostic costs five seconds: compute sin(30). If the answer is 0.5, you are in degrees. If it is −0.988, the machine read 30 radians. If it is 0.454, you have found GRAD mode, probably not on purpose. In code, the fix is the conversion at the boundary: feed Math.sin() degrees times π/180, and convert back with 180/π when presenting results to humans. Every language ships these constants; the bug is forgetting the boundary exists.
Minutes, seconds, and slopes
- Arcminutes and arcseconds subdivide the degree by sixties: 1° = 60′ = 3,600″. Navigation and astronomy live here, and the connection runs deep: one arcminute of latitude is one nautical mile, which is why the knot from our speed guide exists at all. GPS coordinates still print in degrees-minutes-seconds alongside decimal degrees.
- Slopes and grades are angles wearing percentages: a grade is rise over run, so a 100% grade is 45°, and a road sign’s 10% grade is a gentle 5.7°. The percentage and the degree diverge fast, an honest 30° slope is already a 58% grade, which is why ski pistes and mountain roads sound tamer in percent.
Like every family on the site, the full unit list lives in the converter, and the system-wide picture in our unit conversion pillar.
Frequently asked questions
Why do mathematicians insist on radians?
Because calculus rewards them: only in radians is the derivative of sin(x) exactly cos(x), with no stray constant. Every formula involving angles and rates, oscillations, waves, rotations, is cleanest in radians, so the entire mathematical stack standardized on them.
Does anyone actually use gradians?
Some surveying and civil engineering traditions, mostly in continental Europe, where decimal right angles genuinely simplify fieldwork. For everyone else, GRAD is the calculator mode you check for when answers look 10% off.
How precise is the 57.3 shortcut?
180/π is 57.29578, so 57.3 carries an error of 0.007%, far below anything a protractor or even most instruments can see. It is one of the best mental constants in all of unit conversion.
Why 360 degrees and not something rounder?
Ancient convenience that proved unbeatable: 360 divides evenly by more small numbers than any nearby value, so common fractions of a circle are whole numbers of degrees. Decimal alternatives like the gradian have been available for two centuries; 360 has shrugged them off.