How to Use the Rules for 30-60-90 Triangles to Calculate Missing Sides and Angles

Welcome to a quick guide on 30 60 90 triangles! The name of this basically means that the 3 angles of the triangle are 30°, 60°, and 90°. Triangles, especially 30-60-90 ones, are shapes that are heavily used in geometry 🧩, so it's important to be familiar with their characteristics and rules. Let's jump 🐰right into it!

Side Length Rules

30-60-90 triangles are special triangles, meaning their side lengths have a consistent ratio. 💯 These side lengths correspond with the triangle's side measures.

x - The side opposite the 30° angle

x√3 - The side opposite the 60° angle

2x - The side opposite the 90° angle (hypotenuse)

Image from Stack Exchange

You might be wondering how you can easily remember the ratio of the triangle's sides as they correspond with the angle measures 🧐 The 30° angle is the smallest angle, so it corresponds with the smallest side, x! Likewise, the 90° angle is the largest angle, and corresponds with the largest side, 2x 😲

Example Question

Let’s try a quick example question to practice calculating side lengths 💃🏼

Given the right triangle above, what are the side lengths of x and y?

The first step is to determine that this is a 30-60=90 triangle 🤓 There’s a 60° angle and a right angle marked on the triangle already, so the last angle has to be 30°. We’re indeed dealing with a 30-60-90 triangle, so let’s move on to solving for the side lengths!

We can start with x. Referencing our rules above, x is the side opposite the 30° angle. We are given that 12 is the side opposite the 60° angle, so we can use that to help us find x. 

Great, we’ve found x! 👍 Now let’s find y. We already have 2 side lengths, so this side should be a breeze. 💨

y is the side opposite the 90° angle, meaning it’s the largest side. We know from our rules that y = 2x. We already have the value of x.

💡 If you’d like more practice, check out more practice questions related to finding side lengths!

30 60 90 Triangles & Equilateral Triangles

Let’s familiarize ourselves with the relationship between 30 60 90 triangles and equilateral triangles 😎

The large triangle shown is an equilateral triangle, with 60° at each corner. As you can see, it’s been divided ✂️ into two 30 60 90 triangles. Each of the sides is the same length (2x).

Area of 30 60 90 Triangles

Okay, let’s take our knowledge of the side lengths a step farther by talking about area! 🔑 As a reminder, here is the equation of the area of a triangle:

Example Question

Here’s an example question that tests you on area. We’ll be practicing applying the knowledge we just learned. Let’s jump into it 🚗

Given the above right triangle, what is its area?

We have a triangle that has a 30° angle marked, verifying that we have the type of triangle 📐 we need. We have a side length of 6 that is across from the 30° angle, meaning that 6 is our smallest side. Let’s find the side marked a. 🔐

The a side is across from the 60°. The expression to find this side is x√3, and because we already have x = 6, simply plug 6 into the expression. Our side is 6√3! 🎉

Let’s find the area. Simply plug our numbers into the area expression given above and solve.

You’ve got it! 🎊 Our area is 18√3 cm^2. Be sure to remember your units, which are squared because we’re solving for the area.

💡 If you’d like to do more practice questions on your own, use this calculator to check your work!

Conclusion

Congratulations! 👏 Give yourself a pat on the back. You’ve made it to the end of this article! 

Hopefully, you should have a better understanding 🧠 of 30 60 90 triangles and their various applications in geometry. This can be a nuanced topic, so be sure to do lots of practice questions and study up 📝 on the 30 60 90 triangle’s side lengths.

Good luck on your study journey, and check out Fiveable for more geometry resources! 🤸‍♀️