Degrees, Radians, and Gradians: Comparing Angle Units
Most trigonometry only ever mentions two units, degrees and radians, but a third has quietly been in use for centuries in surveying and some engineering contexts: the gradian, which splits a right angle into 100 pieces instead of 90. Lining all three up side by side, with real conversions worked out for each, makes it much easier to see what each one is actually doing, and why, of the three, only radians are built from the circle's own geometry rather than from an arbitrary human choice of how finely to slice a turn.
Three ways to cut up a circle
Degrees divide a full turn into 360 equal parts, a round number with many convenient divisors (2, 3, 4, 5, 6, 8, 9, 10, 12...) that made hand calculation easier long before calculators existed. Gradians (also called gons) divide a full turn into 400 equal parts instead, so a right angle is a clean 100 gradians — the appeal being a base-10 relationship to a right angle, convenient for surveying calculations done in a decimal system. Radians are different in kind, not just in scale: one radian is the angle swept out when the arc length along a circle equals its radius, so a full turn is 2π radians — a value derived from the circle's own circumference rather than chosen for numerical convenience.
The conversion factors
All three units describe the same underlying angle, so converting between any two is a fixed multiplication:
- Degrees to radians: multiply by π/180.
- Radians to degrees: multiply by 180/π.
- Degrees to gradians: multiply by 10/9.
- Gradians to degrees: multiply by 9/10.
- Radians to gradians: multiply by 200/π.
- Gradians to radians: multiply by π/200.
Every one of these six factors comes from the same three equivalences: 360° = 2π radians = 400 gradians. Once you know that a full turn equals all three, dividing down to a half-turn, a quarter-turn, or any other fraction gives every other conversion for free.
Deriving the conversion factors from scratch
Rather than memorising six separate multipliers, it is faster to remember one triple equivalence and derive whichever factor you need on the spot: 360° = 2π radians = 400 gradians. Dividing every term by 360 gives 1° = π/180 radians = 10/9 gradians, which is exactly where the degrees-to-radians and degrees-to-gradians factors above come from. Dividing instead by 2π gives 1 radian = 180/π degrees = 200/π gradians, the source of the radians-to-degrees and radians-to-gradians factors. And dividing by 400 gives 1 gradian = 9/10 degree = π/200 radians. All six conversion factors are really the same equivalence, just divided through by a different one of the three totals.
Standard angles in all three units
Seeing the same set of angles across all three columns makes the comparison concrete. The radian values below come from the site's own degrees-to-radians converter; the gradian values follow directly from the exact 10/9 ratio to degrees.
| Degrees | Radians | Gradians |
|---|---|---|
| 0° | 0 | 0 |
| 30° | π/6 ≈ 0.5236 | 33.33 |
| 45° | π/4 ≈ 0.7854 | 50 |
| 60° | π/3 ≈ 1.0472 | 66.67 |
| 90° | π/2 ≈ 1.5708 | 100 |
| 180° | π ≈ 3.1416 | 200 |
| 270° | 3π/2 ≈ 4.7124 | 300 |
| 360° | 2π ≈ 6.2832 | 400 |
Notice that 90° lands on a perfectly round 100 gradians, which is the whole design goal of the unit — but 30° and 60°, two of the most common angles in the subject, land on repeating decimals (33.33 and 66.67 gradians) that are no friendlier than their degree equivalents. Gradians only pay off cleanly for angles that are already multiples of a right angle; everywhere else, they are just a different awkward number.
Worked conversion: an arbitrary angle
Take 50°, an angle with no particularly clean value in any unit. Running it through the degrees-to-radians converter gives 0.8727 radians (matching 50 × π/180 ≈ 0.8727). Converting the same angle to gradians is a single multiplication: 50 × 10/9 ≈ 55.56 gradians. All three numbers, 50°, 0.8727 rad, and 55.56 grad, describe the exact same rotation — one seventh of a full turn, give or take rounding — just measured on three different rulers.
Worked conversion: starting from gradians
Going the other direction, suppose a surveying instrument reports an angle of 120 gradians. Converting to degrees: 120 × 9/10 = 108° exactly. Feeding 108° into the degrees-to-radians converter gives 1.885 radians, quadrant II, with a reference angle of 72° and sin 108° ≈ 0.9511, cos 108° ≈ −0.309. The gradian measurement was just as valid a starting point as degrees or radians would have been — trigonometric functions do not care which unit handed them the angle, only that the angle gets converted to radians correctly before any sine, cosine, or tangent is evaluated, since that is the unit every standard math library actually computes with internally.
An angle beyond a single turn, in all three units
Nothing about the conversion factors changes once an angle exceeds 360°, since multiplication by a fixed constant works the same regardless of size. Take 400°, a full turn plus an extra 40°. In radians, 400 × π/180 ≈ 6.9813, and normalising that by subtracting a full 2π ≈ 6.2832 leaves ≈ 0.6981 radians, matching 40° converted directly — exactly as expected, since 400° and 40° point in the same direction. In gradians, 400 × 10/9 ≈ 444.44, and subtracting a full 400-gradian turn leaves 44.44 gradians, which is 40° × 10/9 = 44.44 gradians as well. All three units wrap around at the same physical point — a full turn — they just call that point by a different number, which is exactly why normalising an angle (folding it back into a single 0-to-360°, 0-to-2π, or 0-to-400-gradian range) is unit-independent: whichever unit you normalise in, you land on the same physical direction.
A negative angle, in all three units at once
Negative angles convert exactly like positive ones — the sign just carries straight through the multiplication. Take −200°. In radians, −200 × π/180 ≈ −3.4907, and normalising by adding a full 2π turn lands it at 2.7925 radians, matching a positive angle of 160° converted directly (−200° and 160° point in the same direction, one full turn apart). In gradians, −200 × 10/9 ≈ −222.22, and adding a full 400-gradian turn gives 177.78 gradians, matching 160° × 10/9 ≈ 177.78 as well. The degrees-to-radians converter confirms the normalised picture directly: −200° lands in quadrant II with a reference angle of 20°, sin ≈ 0.342, and cos ≈ −0.9397 — exactly the values a straightforward 160° angle would produce, since that is genuinely where −200° points once it's wrapped into a single turn.
Where gradians still show up
Gradians never displaced degrees or radians in mainstream mathematics, but they didn't disappear either. Many scientific calculators still ship with a three-way DEG/RAD/GRAD mode switch alongside degrees and radians, a legacy of the unit's use in continental European surveying and some artillery and engineering calculations, where a decimal-friendly right angle was worth more than compatibility with the 360° convention everyone else used. If you ever see a calculator with a mode you don't recognise labelled "GRAD," this is what it means — and switching into it by accident, then forgetting to switch back, produces exactly the same kind of wildly wrong trig output as a degrees/radians mix-up does.
Why radians are still "the" unit mathematically
Degrees and gradians are both perfectly serviceable for reporting an angle to a person, but neither is built from anything more fundamental than "pick a convenient number of slices." Radians are different because the number 2π is not a choice at all — it is forced by the ratio between a circle's circumference and its radius, a fact about circles that exists independently of any human convention. That is precisely why formulas like arc length s = rθ and sector area ½r2θ only work cleanly in radians: they are built directly from the same geometric relationship that defines the radian in the first place. Swap in degrees or gradians and an extra conversion constant has to be bolted on to make the formula balance, whereas radians need nothing extra because the unit itself already encodes the geometry. What Are Radians, and Why Mathematicians Prefer Them goes deeper into that arc-length definition and why calculus specifically insists on radians rather than either alternative.
Choosing a unit in practice
There is no mathematically "wrong" unit among the three — a triangle solved in gradians and converted back to degrees at the end gives exactly the same answer as one solved in degrees throughout, since a valid conversion never loses or distorts information. The only real risk is switching units partway through a calculation without converting, which corrupts a result just as thoroughly as any other unit mismatch would. In everyday use, reach for degrees when communicating with a person — a compass bearing, a ramp's incline, a rotation instruction — since 360 divides evenly into the fractions people think in (a quarter turn, a third of a circle, and so on). Gradians survive mainly in specific surveying and some European engineering contexts, where the base-10 relationship to a right angle lines up conveniently with decimal distance measurements. Reach for radians the moment calculus, physics, or any programming language's math library gets involved, since Math.sin, Math.cos, and every other standard trig function assume radian input by default — passing a raw degree value into one of them without converting first is one of the most common and easiest-to-miss bugs in code that does any trigonometry at all.
Convert confidently between degrees and radians with the degrees to radians converter, and see the Trig Values Reference for the standard angles above laid out with their full set of trig function values as well. Whichever unit a problem hands you, converting it correctly before doing any trigonometry is always the first step, and never one worth skipping, however tempting it is to assume the units will simply sort themselves out.