The 400-Grad Circle That Should Have Been
In 1793, as the French National Convention was busy inventing the meter, the kilogram, and the liter, a committee including Lagrange, Laplace, Monge, and Condorcet turned their attention to angles. The problem: the degree (1/360 of a circle) was a Babylonian relic based on base-60 counting, incompatible with the new decimal metric system. Their solution: define a new unit where a right angle equals 100 grads, making a full circle 400 grads. Every quadrant would be a clean 100. Every percent slope could be expressed directly in grads without conversion.
The proposal made intellectual sense. 400 is divisible by 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200 — more divisors than 360, which was the main selling point. Navigation charts could use grads directly for compass bearings. Surveyors could calculate slopes with integer arithmetic. Artillery officers could compute elevation corrections with simpler tables.
It didn't matter. By the time the gradian was standardized (in the late 19th century), degrees had colonized astronomy, navigation, mechanical engineering, and education worldwide. Radians — the natural unit for mathematics (where the arc length equals radius times angle in radians) — had captured pure science. The gradian arrived too late for the party that mattered and found refuge in specific niches where its decimal advantages outweighed the cost of being different: geodetic surveying in continental Europe and some fields of civil engineering.
The Conversion Chain: Gon, Deg, Rad
All three units measure the same physical quantity (plane angle) with different scales:
| Quantity | Degrees (°) | Gradians (gon) | Radians (rad) |
|---|---|---|---|
| Full turn | 360 | 400 | 2π ≈ 6.28319 |
| Right angle | 90 | 100 | π/2 ≈ 1.57080 |
| Straight line | 180 | 200 | π ≈ 3.14159 |
| 1 grad/gon | 0.9 | 1 | π/200 ≈ 0.01571 |
| 1 degree | 1 | 1.111... | π/180 ≈ 0.01745 |
| 1 radian | 57.2958 | 63.6620 | 1 |
degrees = gradians × 0.9 radians = gradians × (π / 200)
Worked Examples
100 grads → 90°
A perfect right angle. This is why gradians were appealing: 100 grads = 90° = quarter-turn. In surveying, this means a 100-gon deflection is exactly perpendicular to the baseline — cleaner than saying "89.9997°" when your theodolite reads 100.000 gon.
200 grads → 180°
A straight line. Half-circle. Sunrise to sunset at the equinox (ignoring atmospheric refraction and solar diameter, each of which adds ~0.83° to the visible horizon).
50 grads → 45°
Half a right angle. The pitch of a standard residential roof (often 45°, sometimes 30° or 22.5°). Also the angle of the isosceles triangle formed by cutting a square along its diagonal. In gradians: 50 gon. Clean integer. One reason roofers in France might prefer gons over degrees — though most just use a speed square.
400 grads → 360°
Full circle. The complete rotation. 400 gon = 360° = 2π rad. All three represent the same thing. The difference is which number you prefer to say out loud. Engineers usually pick whichever matches their tool's display mode.
Common Gradian Conversions
| Gradians | Degrees | Radians | Real-world Reference |
|---|---|---|---|
| 1 gon | 0.9° | 0.0157 rad | About 0.5° less than one full moon's apparent diameter. |
| 10 gon | 9° | 0.157 rad | Roughly 18 moon widths across the sky. |
| 50 gon | 45° | 0.785 rad | Roof pitch. Diagonal of a square. Octagon interior angle. |
| 100 gon | 90° | 1.571 rad | Right angle. Corner of your room. |
| 150 gon | 135° | 2.356 rad | Interior angle of a regular octagon. |
| 200 gon | 180° | 3.142 rad | Straight line. Half-circle. Sunset direction at noon. |
| 300 gon | 270° | 5.236 rad | Three-quarter turn. Clock position at 9 o'clock. |
| 400 gon | 360° | 6.283 rad | Full circle. Back where you started. |
Engineering Context
In geomatics and land surveying, the gradian (called "gon" in ISO standards since 1999) remains common for expressing slopes, directions, and angular measurements in geographic information systems (GIS). Many European cadastral systems store angles in gon internally. When importing such data into software that expects degrees (most CAD tools, game engines, CSS transforms), a multiply-by-0.9 conversion is required. Failing to detect the unit produces systematic errors: a 100-gon bearing imported as 100° points 10° off-target — enough to miss a property corner by meters at survey distances. The Mars Climate Orbiter disaster (1999) famously resulted from a unit mix-up between metric and imperial, and while that case involved thrust force rather than angles, it illustrates the general principle: silently wrong units in engineering data destroy hardware before anyone notices. Always verify the angular unit column in your coordinate file header. For related conversions: degrees to radians, radians to degrees, and the comprehensive Speed Conversion Guide.
More: degrees to radians · radians to degrees · Angle Guide
Related Unit Converters
Frequently Asked Questions
Why is my calculator giving weird sine values?
Check the angle mode. If it says "GRAD" or "GON" instead of "DEG," your calculator interprets all inputs as gradians. sin(90) in DEG mode = 1.000. sin(90) in GRAD mode = sin(81°) = 0.9877. Close but wrong. cos(100) in DEG = -0.1736. cos(100) in GRAD = cos(90°) = 0.0000. Very wrong. If your exam answers are consistently off by small amounts, check the DRG key. This single mistake accounts for more lost points on engineering calculus exams than actual conceptual errors.
Are gradians and grades the same thing?
"Grade" can mean several things: a mark in school, the slope of a road (percentage grade), or an angle unit (archaic term for gradian). In modern ISO terminology, the correct name for the metric angle unit is "gon" (plural: "gons"), not "grade." The abbreviation "grad" is deprecated to avoid confusion with other meanings. However, many calculators still label the mode "GRAD," and many engineers still say "grads" colloquially. Context disambiguates.
Do GPS coordinates use degrees or gradians?
Decimal degrees, always. Latitude and longitude are expressed in degrees (DD.DDDDDD format). WGS84, the datum used by GPS, defines coordinates in decimal degrees. Internal computations may use radians (for distance formulas like haversine), but external representation is always decimal degrees. No major GIS format uses gradians for coordinate storage — not Shapefile, not GeoJSON, not KML. If you encounter gradians in coordinate data, someone made a mistake converting from a European survey source that used gon. Multiply by 0.9 and proceed with caution.
What is a centigrad?
An older synonym for gradian (from Latin: centi = hundred + gradus = step). 100 centigrads = 100 grads = 90 degrees. The term fell out of use partly because "centigrade" also referred to the Celsius temperature scale (now officially "Celsius" to avoid confusion), creating ambiguity. Modern usage prefers "gon" or "gradian." Avoid "centigrad" unless you enjoy explaining yourself twice per sentence.