# 30 60 90 Special Triangles: Geometry

Hi everyone and welcome to MathSux! In this post we are going to break down 30 60 90 degree special triangles. What is it? Where did it come from? What are the ratios of it’s side lengths and how to do we use them? You will find all of the answers to these questions below. Also, don’t forget to check out the video below and practice questions at the end of this post. Happy calculating! đź™‚

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## What is a 30 60 90 Triangle and why is it â€śSpecialâ€ť?

The 30 60 90 triangle is special because it forms an equilateral triangle when a mirror image of itself is drawn, meaning all sides are equal!  This allows us to find the ratio between each side of the triangle by using the Pythagorean theorem. Check it out below!

Now letâ€™s draw a mirror image of our triangle.  Next, we can label the length of the new side opposite 30Âş “a,” and add this new mirror image length with the original we had to get, a+a=2a.

If we look at our original 30 60 90 triangle, we now have the following values for each side based on our equilateral triangle:

Now we can re-label our triangle, knowing the length of the hypotenuse in relation to the two legs. This creates a ratio that applies to all 30 60 90 triangles!

## How do I use this ratio?

Knowing the above ratio, allows us to find any length of any and every 30 60 90 triangle, when given the value of one of its sides.

Letâ€™s try an Example:

-> First letâ€™s look at our ratio and compare it to our given triangle.

->Notice we are given the value of a, which equals 4, knowing this we can now fill in each length of our triangle based on the ratio of a 30 60 90 triangle.

Now letâ€™s look at an Example where we are given the length of the hypotenuse and need to find the values of the other two missing sides.

->First letâ€™s look at our ratio and compare it to our given triangle.

-> Notice we are given the value of the hypotenuse, 2a=20. Knowing this we can find the value of a by dividing 20 by 2 to get a=10. Once we have the value of a=10, we can easily find the length of the last leg based on the 30 60 90 ratio:

Now for our last Example, when we are given the side length across from 60Âş and need to find the other two missing sides.

->First letâ€™s look at our ratio and compare it to our given triangle.

-> In this case, we need to use little algebra to find the value of a, using the ratio for 30 60 90 triangles.

Now that we have one piece of the puzzle, the value of a, letâ€™s fill it in our triangle below:

Finally, letâ€™s find the value of the length of the hypotenuse, which is equal to 2a.

## Practice Questions:

Find the value of the missing sides of each 30 60 90 degree triangle.

## Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! đź™‚

Looking to review 45 45 90 degree special triangles? Check out this post here!