# 30-60-90 Triangle

How to solve 30-60-90 triangle problems: definition, property, examples, and their solutions.

## Definition, Property

A 30-60-90 triangle is a triangle

whose interior angles are 30º, 60º, and 90º.

The ratio of its sides is

1 : √3 : 2.

## Example 1

The interior angles are 90º and 30º.

So this is a 30-60-90 triangle.

Draw a 30-60-90 triangle,

whose sides are 1, √3, and 2,

next to the given right triangle.

Then these two triangles are similar.

Since these two triangles are similar,

their sides are proportional.

So 2/1 = *x*/√3.

Similar triangles

Solve the proportion.

Then *x* = 2√3.

Proportion

## Example 2

The interior angles are 60º and 90º.

So this is a 30-60-90 triangle.

Draw a 30-60-90 triangle,

whose sides are 1, √3, and 2,

next to the given right triangle.

Then these two triangles are similar.

Since these two triangles are similar,

their sides are proportional.

So *x*/2 = 5/√3.

Multiply 2 on both sides.

Then *x* = (5/√3)⋅2.

5⋅ = 10

So *x* = 10/√3.

To remove the square root in the denominator,

multiply [√3/√3].

Then *x* = 10√3/3.

Rationalizing the denominator