Radians, degrees, and gradians — when to use which
Why radians are the only honest angle for math, and why gradians exist for surveyors.
Three systems. They all measure the same physical quantity — the angle between two intersecting lines or the rotation of a body.
- Degrees (°) — 360 per full turn. Origin: Babylonian base-60 counting; 360 days in a year. Most familiar. - Radians (rad) — 2π per full turn. The "natural" unit for calculus: sin'(x) = cos(x) only if x is in radians. The SI standard. - Gradians (gon / grad) — 400 per full turn. Each right angle is exactly 100. Used in surveying because right-angle = 100 makes the math integer-clean.
Conversions:
\text{rad} = \text{°} \times \frac{\pi}{180} \text{°} = \text{rad} \times \frac{180}{\pi} \text{gon} = \text{°} \times \frac{10}{9}$
Sub-units exist for astronomy:
- 1° = 60 arcminutes (′) - 1′ = 60 arcseconds (″)
A quarter at 50 km distance subtends about 1.4 milliarcseconds (the size of a coin seen from 4,000 km). That's how precision telescope pointing works.
For your day-to-day (compass, slice of pizza, turn of a steering wheel), degrees are fine. For math, use radians. For civil engineering, gradians. The site supports all of them.