30-60-90 Triangle
TrigonometryA 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 $90^\circ$. The sides always follow a fixed pattern: the shortest side is $1$, the medium side is $\sqrt{3} \approx 1.73$, and the longest side (hypotenuse) is $2$.
Example
If the short side (opposite $30^\circ$) is $4$, then the medium side $= 4\sqrt{3} \approx 6.93$ and the hypotenuse $= 8$.
Key Insight
This triangle is half of an equilateral triangle cut from top to bottom. That is why the short leg is half the hypotenuse.
Definition
A $30$-$60$-$90$ triangle has sides in ratio $1 : \sqrt{3} : 2$. Key trig values: $\sin(30^\circ) = \cos(60^\circ) = 1/2$; $\cos(30^\circ) = \sin(60^\circ) = \sqrt{3}/2$; $\tan(30^\circ) = 1/\sqrt{3} = \sqrt{3}/3$; $\tan(60^\circ) = \sqrt{3}$.
Example
A ladder $12$ m long leans at $60^\circ$ to the ground. Height reached $= 12\sin(60^\circ) = 12(\sqrt{3}/2) = 6\sqrt{3} \approx 10.39$ m. Horizontal reach $= 12\cos(60^\circ) = 12(1/2) = 6$ m.
Key Insight
The side ratio $1:\sqrt{3}:2$ can be verified with the Pythagorean theorem: $1^2 + (\sqrt{3})^2 = 1 + 3 = 4 = 2^2$. Memorizing these ratios makes unit-circle trig values for $30^\circ$ and $60^\circ$ automatic.
Definition
The $30$-$60$-$90$ triangle is half an equilateral triangle (bisected by an altitude). In the unit circle, it corresponds to the angles $\pi/6$ and $\pi/3$ with terminal points $(\sqrt{3}/2, 1/2)$ and $(1/2, \sqrt{3}/2)$. The side ratio $1:\sqrt{3}:2$ is encoded in the algebraic numbers: $\cos(\pi/6) = \sqrt{3}/2$, a value in the cyclotomic field $\mathbb{Q}(\sqrt{3})$.
Example
Regular hexagons are built from $6$ equilateral triangles, each divisible into two $30$-$60$-$90$ triangles. The hexagonal lattice (used in graphene, honeycomb structures) is the geometric consequence of the $30$-$60$-$90$ ratio.
Key Insight
The values $\sin(\pi/3) = \sqrt{3}/2$ and $\cos(\pi/3) = 1/2$ are elements of the cyclotomic field $\mathbb{Q}(e^{2\pi i/12})$. The $30$-$60$-$90$ triangle's side ratios are algebraic integers in this field, connecting elementary geometry to algebraic number theory.