How to solve 30-60-90 triangles

Definition of a 30-60-90 triangles, including angles and side lengths

A $30-60-90$ is a scalene triangle and each side has a different measure.

Since it’s a right triangle, the sides touching the right angle are called the legs of the triangle, it has a long leg and a short leg, and the hypotenuse is the side across from the right angle.

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In this diagram, the short leg is $x$, the long leg is $x\sqrt{3}$, and the hypotenuse is $2x$. These are always the ratios in a $30-60-90$ triangle.

We can use the relationships in the diagram to solve all $30-60-90$ triangles.

How to solve 30-60-90 triangles

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Solving for the side lengths of a 30-60-90 triangle

Example

If $x=6$, what is the length of the hypotenuse and the long leg?

The hypotenuse is related to $x$ by $2x$. We know $x=6$, so the hypotenuse is $2(6)=12$ units. The long leg is related to $x$ by $x\sqrt{3}$, so the long leg is $6\sqrt{3}$ units.

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Let’s look at another example.

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Since it’s a right triangle, the sides touching the right angle are called the legs of the triangle, it has a long leg and a short leg, and the hypotenuse is the side across from the right angle.

Example

What are the lengths of sides $a$ and $b$?

The pattern for the sides of a $30-60-90$ triangle is $x$ for the short leg, $x\sqrt{3}$ for the long leg, and $2x$ for the hypotenuse. In this case we know the hypotenuse is $4\sqrt{3}$, so

$2x=4\sqrt{3}$

$x=2\sqrt{3}$

The short leg $b$ is $2\sqrt{3}$. The long leg $a$ is $x\sqrt{3}$, or

$2\sqrt{3}\sqrt{3}=2(3)=6$

The long leg $a$ is $6$ units.

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Let’s try another example.

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Example

What are the lengths of the missing sides?

We know the long leg is $5\sqrt{3}$, so

$x\sqrt{3}=5\sqrt{3}$

$x=5$ units

The short leg $b$ is $5$ units. Then the hypotenuse is

$2x=2(5)=10$ units

The hypotenuse $c$ is $10$ units.

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