Core answer: A full circle is 360° = 2π radians = 400 gradians. The bridge: 180° = π rad, so 1° = π/180 ≈ 0.01745 rad and 1 rad ≈ 57.3°. Slopes and gradients: a 10% grade = arctan(0.10) ≈ 5.7°; a 45° slope is exactly 100% grade. GPS coordinates split degrees into 60 minutes and 3,600 seconds.
The three systems
| System | Full circle | Used by |
|---|---|---|
| Degrees (°) | 360 | everyone, navigation |
| Radians (rad) | 2π ≈ 6.283 | math, physics, code |
| Gradians (gon) | 400 | surveying (rare) |
Key conversions: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π.
Why radians exist
Radian = angle where arc length equals radius. That makes the beautiful formulas work: arc = rθ, sin θ ≈ θ for small θ, and calculus derivatives clean (d/dx sin x = cos x only in radians). When your calculator computes sin(1) it assumes radians unless told otherwise.
Worked examples
Example 1 — Slope signs. A road sign "10%" means rise/run = 0.10 → 5.7°. Wheelchair ramps require ≤ 1:12 (8.3% = 4.8°). San Francisco's steepest streets hit ~31.5% ≈ 17.5°.
Example 2 — Roof pitch. Chinese roofers say "几分水" — 5分水 ≈ 50% slope ≈ 26.6°; snowy regions build 8–10分水 (39–45°) to shed loads.
Example 3 — GPS coordinates. 39°54'26.4"N = 39 + 54/60 + 26.4/3600 = 39.9073° — Beijing. One second of latitude ≈ 30.9 m on the ground.
Example 4 — Code bug. Math.sin(90°) in most languages returns 0.894, not 1 — because the function expected radians: sin(90 × π/180) = sin(π/2) = 1.
Common mistakes and myths
- DEG/RAD mode chaos — sin(30) = 0.5 in DEG, −0.988 in RAD; 90% of "the calculator is broken" reports are mode errors.
- Percent grade vs degrees — 100% grade = 45°, not 90°; a "vertical" wall is infinite percent.
- Assuming minutes are decimal — 30' = 0.5°, but 45' = 0.75°, not 0.45°; DMS↔decimal conversion errors misplace GPS points by kilometers.
- Forgetting wraparound — bearings jump from 359° to 0°; averaging 350° and 10° as (350+10)/2 = 180° points exactly backwards (correct circular mean ≈ 0°).
- Small-angle approximation abuse — sin θ ≈ θ (radians) is great under 5° (0.4% error) but 8% wrong by 30°.