Brian McCall

**Univ. of Wisconsin**

J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

Share

A 30 60 90 triangle is a special type of right triangle. What is special about **30 60 90 triangles** is that the sides of the 30 60 90 triangle always have the same ratio. Therefore, if we are given one side we are able to easily find the other sides using the ratio of 1:2:square root of three. This special type of right triangle is similar to the 45 45 90 triangle.

There are two special array triangles, 45, 45,

90 and 30, 60, 90 triangles.

Now what's special about this 30, 60, 90.

triangle is that of the three sides that.

are all going to be different, I only

need to know one, and I can find the other two.

So that's pretty amazing, that of the three

sides of a triangle, when it's a

30-degree, a 60-degree and a 90-degree,

I only need to know one side and

I can find the other ones.

So the shortcut, and again this is based

on the Pythagorean theorem, is to find

your longer leg

Now how do I know that. this is my longer leg.

Well, if I look at my angle measures, 30

degrees is my smallest measure, which

means the side that is opposite

of 30 will be my shortest leg.

The next longest will be

my next largest angle.

And since 60 degrees is smaller than 90,

60 degrees is my next longest leg.

And then, of course, our hypotenuse is always

the longest side in our triangle.

So to go from our shorter leg to our longer

leg, I need to multiply by the square

root of 3. So if this side is, let's

say, five units, then your

longer leg will be 5 times the square

root of 3.

Now, let's say you want to find your hypotenuse.

To go from your shorter leg to your hypotenuse,

all you need to do is multiply.

by 2. So this is going to be 2.

times S. So if S is 5, your

hypotenuse will be 10.

So this works in both directions.

So let's say you had your longer leg and

you wanted to find the other two sides.

Well, first -- and actually I forgot to

write down right here that you would

multiply by the square root of 3.

So let's say you wanted to go

in the opposite direction.

From your longer leg to your shorter leg.

Well, the inverse operation of multiplying

by 3 is to divide by the square root.

of 3. The inverse operation of going

from your hypotenuse to your

shorter leg is going to be not multiplying.

by 2 but dividing by 2. So.

so the key leg in this triangle

is this shorter leg.

Because from here you can find your hypotenuse

by multiplying by 2 and you can

find your longer leg.

So use these shortcuts whenever you see.

a triangle with 30, 60 and 90 degrees.