Guide

Angle units in engineering and surveying

Degrees, radians and gon all describe an angle but are made for different purposes. The radian is dimensionless because it is a ratio of lengths — which is why formulas often state angles without a unit at all.

Degrees and radians

A full turn is 360 degrees or 2π radians. So 180 degrees corresponds exactly to π radians, about 3.14159265358979. The radian is defined as the ratio of arc length to radius and is therefore dimensionless: an angle of 1 rad means the arc is exactly as long as the radius.

Gon in surveying

The gon divides a full turn into 400 parts rather than 360. A right angle is therefore exactly 100 gon, which simplifies calculation in surveying. It is common in German-speaking countries as Neugrad and in English as gradian.

Degrees and radians on a calculator

Calculators and spreadsheets have a switch between degrees and radians that easily sits in the wrong position unnoticed. The sine of 90 degrees is 1, while the sine of 90 radians is about 0.894. Because both modes return plausible values, the error often only shows up in the result. Two control values help: sin(30°) = 0.5 and cos(0) = 1.

Arc minute and arc second

A degree is divided into 60 arc minutes and an arc minute into 60 arc seconds. Coordinates are often stated in this form, for example 48° 8′ 30″. One arc minute along a meridian corresponds to about one nautical mile on the Earth surface, that is 1852 metres — this relation is why the nautical mile is used in navigation.

Matching conversions

Matching tables

Unit family: Angle

Sources