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