If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Geometry (all content)

### Course: Geometry (all content)>Unit 13

Lesson 1: Special right triangles

# Special right triangles intro (part 2)

A 30-60-90 triangle is a special right triangle with angles of 30, 60, and 90 degrees. It has properties similar to the 45-45-90 triangle. The side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is the length of the short leg times the square root of three. Alternatively, the ratio of the sides is 1 : √3 : 2. Created by Sal Khan.

## Want to join the conversation?

• Where did he get the 4 at ? And where did he get the 3/4 at ??? • i do not necessarily need the property of the 45 - 45 - 90 triangle because I can always derive it from the pythagorean theorem, right ?? • When Sal goes onto the second special triangle type, 30-60-90 triangles, he gives the equation for when solving B or A using C, in other words, we know the equation for finding the side lengths using the hypotenuse. How do we find the side lengths to find the other side and use it to find the hypotenuse? Is there an equation? • Rather than an equation per se, it is more thinking about ratios, possibly needing to set up an equation to find the answer, but once you learn the pattern, most of the time you will just be able to write the answers.
So the ratio for the 30-60-90 triangle is x, x√3, 2x.
If we have the hypotenuse (lets say 6), then 2x = 6, divide by 2 to get x = 3. The equation will always be the same, so dividing by 2 will always get the side opposite the 30, and to get the side opposite the 60, just tack on √3, answer will be 3√3.
If we have the side opposite the 30 (lets say 3), we double to get the hypotenuse of 6 and tack on √3 so side opposite 60 is 3√3.
The hardest is when the side opposite the 60 is an integer (lets say 9). In this case, we set up an equation x√3 = 9, divide by √3 to get x = 9/√3, we do not like roots in denominator, so multiply by √3/√3 which will end up as 9√3/3 = 3√3 for side opposite the 30, and double to get hypotenuse of 6√3. So the pattern would be multiplying the side opposite the 60 by √3/3 to get the side opposite the 30 which is why we like numbers that are multiples of 3 to go through this process, but it does not have to be.
In 45-45-90 with a ratio of x, x, x√2, if hypotenuse is an integer (say 8), we do the same process of multiplying by √2/2 to get 4√2 for the two legs.
Many students use a visual of a tic-tac-toe board, left side is the angles of a triangle (either 45-45-90 or 30-60-90), write the ratios in the three middle boxes (either x x x√2 or x x√3 2x), then fill in the given side in the opposite angle slot and calculate from there.
So, while you can use equations, it is easier to learn patterns which take only a few seconds to calculate.
• Where does the 4 in h^2/4 () come from? • I understand how the different sides relate to the hypotenuse but how do you find the Hypotenuse from the side opposite the 60 degree angle? • At , I STILL don't understand how you go from subtraction in the previous step (h-squared MINUS h-squared over 4) to multiplication (h-squared TIMES 1 minus 1/4). How is that factoring? Specifically WHICH factoring video explains this? • I'm sorry, I don't have a video for you, but I can explain it.
Try to do the step the other way around:
`h^2 * (1 - 1/4) =h^2 * 1 + h^2 * -1/4 =h^2 - h^2/4`
h^2 is distributed to all elements inside the parentheses. Watch out, though! This distribution only works with subtraction or addition in the parentheses, with division or multiplication, only once is enough, since all factors are anyway distibuted (I like to imagine it them as beign mixed and matched again) in the parentheses.
If you do the step the right way around, you realise that you have to divide h^2 from these elemans to be able to multiply it later within the parentheses:
`h^2 - h^2/4 =h^2 * (h^2/h^2 + (-h^/4)/h^2) =h^2 * (1 - 1/4)`
And you're there!
I hope I could help and that I wasn't too late. :)
• At about minutes into the video, Sal said, he shouldn't have used the A, because it's to do with area or something?
I didn't understand what he meant by that. The A is the length and not the area right? • at how did (5√2)²+ (5√2)²=100   