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Degrees to gradians: exact 10/9 factor, 400-gon circle, conversion chart, and surveying context
Conversion formula
Verification: factors follow standard unit definitions; round for display only.
Quick reference chart
| Degree | Gradian |
|---|---|
| 1 | 1.111111 |
| 2 | 2.222222 |
| 3 | 3.333333 |
| 4 | 4.444444 |
| 5 | 5.555556 |
| 6 | 6.666667 |
Educational explanation
Degrees to gradians
Convert ° to gon (grad) when a CAD title block, compass rose, or trig homework uses degrees but your theodolite readout, European cadastral map, or military grid tool expects gradians.
The gradian (gon, grad) divides a right angle into 100 gon and a full circle into 400 gon. Because 400 gon = 360°, one degree equals 10/9 gon (about 1.111… gon). This site's angle catalog stores Gradian = 0.9 relative to Degree = 1, so:
gon = ° ÷ 0.9 · equivalently gon = ° × 1.11111
The factor is exact by definition (ratio 10/9), not an approximate constant. Divide degrees by 0.9; multiply gradians by 0.9 for the inverse.
Step-by-step conversions (worked examples)
90° (right angle):
- 90 ÷ 0.9 = 100 gon
- Sanity check: 100 gon is a quarter turn in the gon system.
45° (half of a right angle):
- 45 ÷ 0.9 = 50 gon
- Numeric: 50 gon
36° (typical classroom example):
- 36 ÷ 0.9 = 40 gon
- 40 gon is 10% of a full 400-gon circle.
Degree to gradian conversion chart
| Degrees (°) | Gradians (gon) | Typical context |
|---|---|---|
| 1° | 1.11111 gon | Fine adjustment |
| 9° | 10 gon | Clean decimal in gon system |
| 45° | 50 gon | Diagonal, half right angle |
| 90° | 100 gon | Right angle |
| 120° | 133.333 gon | Equilateral triangle interior |
| 180° | 200 gon | Straight angle |
| 270° | 300 gon | Three-quarter turn |
| 360° | 400 gon | Full circle, closed traverse |
Where degrees → gradians comes up
- Surveying & cadastre: US plats and GIS exports use degrees; European total stations often default to gon.
- Military grids: NATO fire-control tables may mix gon with degree bearings—convert before plotting on gon-based charts.
- Education: Gon exercises teach decimal-friendly right angles (100 gon = 90°).
Gradians to degrees
Convert gon to ° when a theodolite readout, European cadastral map, or slope table lists gradians but your CAD dimension or compass bearing expects degrees.
Multiply gradians by 0.9, or divide by 1.11111 (the reciprocal from the catalog).
° = gon × 0.9 · catalog: ° = gon × 0.9
Step-by-step conversion (worked example)
Convert 100 gon to degrees:
- 100 × 0.9 = 90°
- Sanity check: 100 gon is a right angle in the gon system.
Gradian to degree conversion chart
| Gradians (gon) | Degrees (°) | Typical context |
|---|---|---|
| 1 gon | 0.9° | Definition step (0.9°) |
| 50 gon | 45° | Diagonal, 1:1 slope in metric surveying |
| 100 gon | 90° | Right angle (quarter turn) |
| 200 gon | 180° | Straight angle, half turn |
| 400 gon | 360° | Full circle, closed traverse |
See also gradian to degree.
Gradians vs degrees vs radians, and related tools
400 gon = 360° = 2π rad. Use 10/9 for °→gon and 0.9 for gon→°; do not use radian formulas unless you convert through degrees first.
Angle systems at a glance
| Topic | Gradian (gon) | Degree (°) |
|---|---|---|
| Full turn | 400 gon | 360° |
| Right angle | 100 gon | 90° |
| Straight angle | 200 gon | 180° |
| 1° in the other unit | 1.11111 gon | 1° |
| Primary use | Surveying (EU), metric civil | Global CAD, navigation, everyday |
Common mistakes
- Using 100 gon = 180° by mistake—100 gon is 90°; 200 gon is 180°.
- Applying π/180 directly to gon—gon uses 0.9, not π.
- Mixing percent slope with gon—slope percent is rise/run, not an angle unit.
Related angle converters
Gradian to degree, degrees to radians, radians to degrees, and degree to arc minute.
Frequently asked questions
How do you convert degrees to gradians?
Divide degrees by 0.9: gon = ° ÷ 0.9. Equivalently, gon = ° × 1.11111 (10/9). Example: 90° → 100 gon.
How do you convert gradians to degrees?
Multiply gradians by 0.9: ° = gon × 0.9.
How many gradians is 45 degrees?
45 ÷ 0.9 = 50 gon.
How many gradians is 90 degrees?
90 ÷ 0.9 = 100 gon—a right angle in the gon system.
How many gradians is 360 degrees?
360 ÷ 0.9 = 400 gon—a full circle.
What is 1 degree in gradians?
1° = 1.11111 gon exactly (10/9 gon). Catalog: Degree = 1, Gradian = 0.9.
Why are there 400 gradians in a circle?
The gradian system divides a right angle into 100 equal parts, so four right angles yield 400 gon per full turn—decimal-friendly for surveying.
Is gradian the same as radian?
No. Radians use arc length on a unit circle (2π per turn). Gradians use 400 per turn with 1 gon = 0.9°.
Is the 0.9 / 10/9 factor exact?
Yes. It follows from 400 gon = 360°, so 1 gon = 360/400 = 0.9° and 1° = 400/360 = 10/9 gon.
How many degrees is 200 gradians?
200 × 0.9 = 180°—a straight angle.
Can I round-trip degrees and gradians?
Yes. Convert 36° → 40 gon → 36° within floating-point limits. Round only at the final display step.
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