Introduction to 30-60-90 Triangle Formula

A 30-60-90 triangle is a special type of right triangle that has angle measures of 30°, 60°, and 90°.

30-60-90 Triangle Formula

What makes this triangle unique is that the sides of the triangle follow a consistent ratio, allowing us to easily calculate the lengths of any side if we know one of them.

In this tutorial, we’ll cover the characteristics of a 30-60-90 triangle, the formula to calculate the sides, and how to apply it in various examples.

Properties of a 30-60-90 Triangle

In a 30-60-90 triangle:

  • The 90° angle is the right angle.
  • The 30° angle is the smallest angle.
  • The 60° angle is the second largest angle.
  • The sides of the triangle follow a specific ratio based on these angles.

Side Length Ratios

The sides of a 30-60-90 triangle are always in the ratio:

1:3:2

30-60-90 Triangle with Side Lengths

Here’s how the ratio corresponds to the sides of the triangle:

  • The shortest side (opposite the 30° angle) is x, which is height in the above diagram.
  • The longer leg (opposite the 60° angle) is x3, which is base in the above diagram.
  • The hypotenuse (opposite the 90° angle) is 2x.

Knowing this ratio, you can determine the length of any side of the triangle as long as you know one of the sides.

Formula for the 30-60-90 Triangle

If you know one side of a 30-60-90 triangle, you can use the following formulas to find the other sides:

If you know the shorter leg (opposite the 30° angle):

Longer leg=x3

Hypotenuse=2x

If you know the longer leg (opposite the 60° angle):

Shorter leg=Longer leg3

Hypotenuse=2×(Longer leg3)

If you know the hypotenuse:

Shorter leg=Hypotenuse2

Longer leg=Hypotenuse2×3

Example Calculations

Let’s look at some example calculations to understand how these formulas work.

Example 1: Finding the longer leg and hypotenuse

Given a 30-60-90 triangle where the shorter leg (opposite the 30° angle) is 5 cm, we can calculate the lengths of the other sides:

30-60-90 Triangle - Example 1

Using the formula for the longer leg:

Longer leg=5×35×1.732=8.66cm

Using the formula for the hypotenuse:

Hypotenuse=2×5=10cm

30-60-90 Triangle - Example 1 Answer

Example 2: Finding the shorter leg and hypotenuse

If the longer leg (opposite the 60° angle) is 6 cm, we can find the shorter leg and the hypotenuse:

30-60-90 Triangle - Example 2

Using the formula for the shorter leg:

Shorter leg=6361.732=3.46cm

Using the formula for the hypotenuse:

Hypotenuse=2×3.46=6.92cm

30-60-90 Triangle - Example 2 Answer