Degrees, radians and gradians are all different ways of measuring angles, and there isn't a standard, they all have their uses and so all of them are used.

## Degrees

Degrees are denoted by the symbol: °.

Degrees measure angles where a right-angle is 90°, this means that a line has an angle of 180° and that a circle has an angle of 360°.

Radians can be denoted by the symbol: r, but often no symbol is used.

The greek letter pi (π) has been used as a constant of the ratio of a circle's circumference to its diameter. ${\displaystyle \pi \approx 3.1415926536}$ .

Radians measure angles where a right-angle is ${\displaystyle {\frac {\pi }{2}}}$ , this means that a line has an angle of ${\displaystyle \pi }$  and that a circle has an angle of ${\displaystyle 2\pi }$ .

As ${\displaystyle 2\pi }$  is a bit of a weird number for a full circle, the greek letter tau (τ) is often used to mean ${\displaystyle 2\pi }$ [1]. ${\displaystyle \tau \approx 6.2831853072}$ . The benefit of using this new constant is that now a right angle (a quarter of a circle) is ${\displaystyle {\frac {\tau }{4}}}$ , half of the circle is ${\displaystyle {\frac {\tau }{2}}}$ , three-quarters of a circle is ${\displaystyle {\frac {3\tau }{4}}}$  and a full circle is ${\displaystyle \tau }$ . As this isn't the standard, throughout this book π will be used instead.

Gradians are denoted by the symbol: g.

Gradians are only used in continental Europe[2].

Gradians measure angles where a right-angle is 100g, this means that a line has an angle of 200g and that a circle has an angle of 400g.

## Converting between them

${\displaystyle x^{\circ }}$  ${\displaystyle x{\frac {\pi }{180}}}$  ${\displaystyle {x{\frac {10}{9}}}^{g}}$
${\displaystyle {x{\frac {180}{\pi }}}^{\circ }}$  ${\displaystyle x}$  ${\displaystyle {x{\frac {200}{\pi }}}^{g}}$
${\displaystyle {x{\frac {9}{10}}}^{\circ }}$  ${\displaystyle x{\frac {\pi }{200}}}$  ${\displaystyle x^{g}}$
${\displaystyle {\frac {\pi }{180}}}$  1.111...g
30° ${\displaystyle {\frac {\pi }{6}}}$  33.333...g
45° ${\displaystyle {\frac {\pi }{4}}}$  50g
60° ${\displaystyle {\frac {\pi }{3}}}$  66.666...g
90° ${\displaystyle {\frac {\pi }{2}}}$  100g
180° ${\displaystyle \pi }$  200g
270° ${\displaystyle {\frac {3\pi }{2}}}$  300g
360° ${\displaystyle 2\pi }$  400g