30-60-90 Triangle
A 30-60-90 triangle is a special right triangle where the sides are in the ratio $1 : \sqrt{3} : 2$.
Formula
\text{sides: } a, a\sqrt{3}, 2a \text{ (opposite } 30^\circ, 60^\circ, 90^\circ \text{ respectively)}
Definition
A $30$-$60$-$90$ triangle has angles of $30^\circ$, $60^\circ$, and $90^\circ$ with sides always in a fixed ratio, $1 : \sqrt{3} : 2$: the shortest side is $1$, the medium side is $\sqrt{3} \approx 1.73$, and the hypotenuse is $2$. Key trig values: $\sin(30^\circ) = \cos(60^\circ) = 1/2$; $\cos(30^\circ) = \sin(60^\circ) = \sqrt{3}/2$; $\tan(30^\circ) = \sqrt{3}/3$; $\tan(60^\circ) = \sqrt{3}$. On the unit circle, it corresponds to the angles $\pi/6$ and $\pi/3$ with terminal points $(\sqrt{3}/2, 1/2)$ and $(1/2, \sqrt{3}/2)$, and the side ratio is encoded in algebraic numbers in the cyclotomic field $\mathbb{Q}(\sqrt{3})$.
Example
If the short side (opposite $30^\circ$) is $4$, the medium side $= 4\sqrt{3} \approx 6.93$ and the hypotenuse $= 8$. A ladder $12$ m long leaning at $60^\circ$ reaches a height of $12\sin(60^\circ) = 6\sqrt{3} \approx 10.39$ m with a horizontal reach of $12\cos(60^\circ) = 6$ m. Regular hexagons are built from $6$ equilateral triangles, each divisible into two $30$-$60$-$90$ triangles, and the hexagonal lattice used in graphene and honeycomb structures is a geometric consequence of this ratio.
Key Insight
This triangle is half of an equilateral triangle cut from top to bottom, which is why the short leg is half the hypotenuse, and the ratio $1:\sqrt{3}:2$ can be verified with the Pythagorean theorem: $1^2 + (\sqrt{3})^2 = 4 = 2^2$. The values $\sin(\pi/3) = \sqrt{3}/2$ and $\cos(\pi/3) = 1/2$ are algebraic integers in the cyclotomic field $\mathbb{Q}(e^{2\pi i/12})$, connecting elementary geometry to algebraic number theory.