30-60-90 triangle

A 30-60-90 triangle is a right triangle with a 30 degree angle, a 60 degree angle,a 90 degree angle. It is special because it side lengths are always in the same ratio. The length of the hypotenuse is twice the length of the shorter leg and the length of the longer leg is the length of the shorter leg.

Simply put, the ratio in order of the sides opposite to the angles for any value of is  :  : . It is a simple "pneumonic" ratio that will help you remember the ratio and each part corresponds to the part of the 30-60-90 triangle. (For example: the corresponds to the angle)

Proofs

Consider a right triangle with angles \(30^\circ\), \(60^\circ\), and \(90^\circ\). Let:

• \(a\): side opposite the \(30^\circ\) angle
• \(b\): side opposite the \(60^\circ\) angle
• \(c\): hypotenuse

Using trigonometric identities

1. Hypotenuse

Here are two pieces of evidence:

\(\sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{AB}{AC}\)

\(\sin(30^\circ) = \frac{1}{2}\)

Setting them equal to each other, we have:

2. Side

\(\cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{BC}{AC}\)

Since \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\):

In conclusion, the side lengths are:

\begin{array}{l} \text{Opposite } 30^\circ: AB \\ \text{Opposite } 60^\circ: AB\sqrt{3} \\ \text{Hypotenuse }  : 2AB \end{array}

Therefore, the side ratios are

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