How To Solve Special Right Triangles

Having trouble with this one?  Take a look.  Email me if you get stuck.  We’ll get you unstuck!

30-60-90 Triangles

LABEL YOUR TRIANGLE FIRSTDiagram of right triangle

The side opposite the 30 degree angle is always the shortest side. Let’s start there. The shortest side is always half of the hypotenuse…or you could say the hypotenuse is two times the short side. The hypotenuse is the longest side. That leaves the side opposite the 60 degree angle….the middle side. It is always √3 times the short side. In other words:

Short Side = x
Medium Side = √3 x or x√3         (whichever way you like to see it)
Long Side = 2x
45-45-90 Triangles

The two sides opposite the 45 degree angles are always congruent. The third side is the hypotenuse.  It is always √2 times the shorter sides. 

Both Congruent Sides = x
Hypotenuse = x √2    or    √2x           ( whichever way you like to write it)

Sometimes your teacher may ask you to find the actual length of each side.  If so, just use your calculator to multiply by √2.


It’s easiest to do IF you have the length of the short side.  If you don’t, then it is the first thing you should find.  You can do this by dividing the hypotenuse by √2 in a 45-45-90 triangle. 

In a 30-60-90 triangle, divide the hypotenuse by 2 to get the shortest side.   If you don’t have the hypotenuse either (yuk!), then you need another way to find the shortest side.  So you can divide the longer leg (of a 30-60-90) by √3.  It’s always a little trickier to “go backwards” and divide to get back to the short side, but it’s doable.

Need help?  Email me.


3 responses to “How To Solve Special Right Triangles

  1. my problem is when the 306090 and 454590 are connected as one back to back

  2. Im like so confused about this stuff and i dnt get it. i just learned this today in my 1st block and is crazy hard

    • Okay. It’s not so hard. Get this part first. If you have a triangle with angles 30, 60, and 90 degrees, it always goes like this: Put your finger on the 30 degree angle. The side directly across from it is the shortest side always. Now put your finger on the 90 degree angle. The side directly across from this angle is always two times as long as the shortest side. Sounds hard. It’s not. Get this part down first and then proceed. But this is where you must start.

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