# The 30-60-90 Triangle

A 30-60-90 triangle is a special right triangle. The other type of special right triangle is 45-45-90. These numbers represent the degree measures of the angles.

 The reason these triangles are considered special is because of the ratios of their sides - they are always the same! Special right triangles hold many applications in both geometry and trigonometry. In this lesson you will learn the general formula for the ratios, and how to find missing sides of any 30 60 90 right triangle. We even offer a free study guide that easy explains the steps easily on one page!
 The General Formula of a 30-60-90 Triangle

 Giving the Sides their Special Names

Before we explain how to find the missing sides, we need to give specific names to each of the sides. For example, the side across from the right angle is always called the hypotenuse (you probably already knew that from the pythagorean theorem). See below.

The side across from the 30° angle is known as the short leg (S.L.).

The side across from the 60° angle is known as the long leg (L.L.).

The side across from the 90° angle is known as the hypotenuse (Hyp.).

Important: Regardless of what direction the triangle is turned, the sides are still named according to their relationship to the angles.

Look at the examples below: The sides are always across from their angle

 How to Find the Missing Side of a 30-60-90 Triangle

You will always be given one of the three sides. What you do from there depends on which side that you are given. Consider each case below.

 Case #1: I know the Short Leg Find Hyp: Multiply by 2 Find L.L: Multiply by Sqrt(3) Case #2: I know the Hypotenuse Find S.L: Divide by 2 Find L.L. Use Case #1 Case #3: I know the Long Leg Find S.L: Divide by Sqrt(3) Find Hyp: Use Case #1 Note: Always find S.L. first and then use Case #1!

 Examples: All Three Cases

Case #1: I know the Short Leg

Case #2: I know the Hypotenuse

Case #3: I know the Long Leg