Guide to Game Development/Theory/Mathematics/Trigonometry/Degrees Vs Radians Vs Gradians

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

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. $\pi \approx 3.1415926536$.

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

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

Gradians

Gradians are denoted by the symbol: ^g.

Gradians are only used in continental Europe.

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

Converting between them

Degrees	Radians	Gradians
$x^{\circ }$	$x{\frac {\pi }{180}}$	${x{\frac {10}{9}}}^{g}$
${x{\frac {180}{\pi }}}^{\circ }$	$x$	${x{\frac {200}{\pi }}}^{g}$
${x{\frac {9}{10}}}^{\circ }$	$x{\frac {\pi }{200}}$	$x^{g}$
1°	${\frac {\pi }{180}}$	1.111...g
30°	${\frac {\pi }{6}}$	33.333...g
45°	${\frac {\pi }{4}}$	50g
60°	${\frac {\pi }{3}}$	66.666...g
90°	${\frac {\pi }{2}}$	100g
180°	$\pi$	200g
270°	${\frac {3\pi }{2}}$	300g
360°	$2\pi$	400g