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## Geometry (all content)

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

Lesson 1: Special right triangles- Special right triangles intro (part 1)
- Special right triangles intro (part 2)
- 30-60-90 triangle example problem
- Special right triangles
- Special right triangles proof (part 1)
- Special right triangles proof (part 2)
- Area of a regular hexagon
- Special right triangles review

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# 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 at7:46? And where did he get the 3/4 at8:12???(7 votes)
- i do not necessarily need the property of the 45 - 45 - 90 triangle because I can always derive it from the pythagorean theorem, right ??(8 votes)
- Yes, but the property gets it much faster.(4 votes)

- 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?(3 votes)
- 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.(4 votes)

- Where does the 4 in h^2/4 (7:47) come from?(3 votes)
- Before that, Sal is squaring (1/2 h)^2. This can be rewritten as (h/2)^2, so squaring the numerator and denominator gives h^2/4.(4 votes)

- 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?(2 votes)
- You need to find the shorter leg first (the one opposite the 30 degree angle) by dividing the longer leg by the square root of 3. Then find the hypotenuse by multiplying the shorter leg by 2.(3 votes)

- At8:03, 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?(2 votes)
- 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. :)(3 votes)

- At about8:40minutes 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?(3 votes)- Yes, A is the length not the area of the triangle.(1 vote)

- at2:03how did (5√2)²+ (5√2)²=100(2 votes)
- so one side is A, another side is B, and the hypotenuse is C, correct? He calls c "h", and the two other sides are 1/2(h), and the other side ((√ 3)/2)h, does it matter which one is which or does the side opposite to the 30º always have to be 1/2(h), and the side opposite to 60º angle always have to be ((√ 3)/2)h?(2 votes)
- It has to always stay that way. You can see why when you compare the right-angled triangle (3:30) with the equilateral triangle (4:20) you get after mirroring it.(2 votes)

- at2:03how did (5√2)+ (5√2)=100(2 votes)

## Video transcript

Sorry for starting the
presentation with a cough. I think I still have a little
bit of a bug going around. But now I want to continue
with the 45-45-90 triangles. So in the last presentation we
learned that either side of a 45-45-90 triangle that isn't
the hypotenuse is equal to the square route of 2 over 2
times the hypotenuse. Let's do a couple
of more problems. So if I were to tell you that
the hypotenuse of this triangle-- once again,
this only works for 45-45-90 triangles. And if I just draw one 45
you know the other angle's got to be 45 as well. If I told you that the
hypotenuse here is, let me say, 10. We know this is a hypotenuse
because it's opposite the right angle. And then I would ask you
what this side is, x. Well we know that x is equal
to the square root of 2 over 2 times the hypotenuse. So that's square root
of 2 over 2 times 10. Or, x is equal to 5
square roots of 2. Right? 10 divided by 2. So x is equal to 5
square roots of 2. And we know that this side
and this side are equal. Right? I guess we know this is an
isosceles triangle because these two angles are the same. So we also that this
side is 5 over 2. And if you're not
sure, try it out. Let's try the
Pythagorean theorem. We know from the Pythagorean
theorem that 5 root 2 squared, plus 5 root 2 squared is equal
to the hypotenuse squared, where the hypotenuse is 10. Is equal to 100. Or this is just 25 times 2. So that's 50. But this is 100 up here. Is equal to 100. And we know, of course,
that this is true. So it worked. We proved it using the
Pythagorean theorem, and that's actually how we
came up with this formula in the first place. Maybe you want to go back to
one of those presentations if you forget how we
came up with this. I'm actually now going
to introduce another type of triangle. And I'm going to do it the same
way, by just posing a problem to you and then using the
Pythagorean theorem to figure it out. This is another type
of triangle called a 30-60-90 triangle. And if I don't have time
for this I will do another presentation. Let's say I have a
right triangle. That's not a pretty one,
but we use what we have. That's a right angle. And if I were to tell you that
this is a 30 degree angle. Well we know that the
angles in a triangle have to add up to 180. So if this is 30, this is 90,
and let's say that this is x. x plus 30 plus 90 is equal to
180, because the angles in a triangle add up to 180. We know that x is equal to 60. Right? So this angle is 60. And this is why it's called a
30-60-90 triangle-- because that's the names of the three
angles in the triangle. And if I were to tell you that
the hypotenuse is-- instead of calling it c, like we always
do, let's call it h-- and I want to figure out the other
sides, how do we do that? Well we can do that
using pretty much the Pythagorean theorem. And here I'm going to
do a little trick. Let's draw another copy of this
triangle, but flip it over draw it the other side. And this is the same triangle,
it's just facing the other direction. Right? If this is 90 degrees
we know that these two angles are supplementary. You might want to review the
angles module if you forget that two angles that share kind
of this common line would add up to 180 degrees. So this is 90, this
will also be 90. And you can eyeball it. It makes sense. And since we flip it, this
triangle is the exact same triangle as this. It's just flipped
over the other side. We also know that this
angle is 30 degrees. And we also know that this
angle is 60 degrees. Right? Well if this angle is 30
degrees and this angle is 30 degrees, we also know that this
larger angle-- goes all the way from here to here--
is 60 degrees. Right? Well if this angle is 60
degrees, this top angle is 60 degrees, and this angle on the
right is 60 degrees, then we know from the theorem that we
learned when we did 45-45-90 triangles that if these two
angles are the same then the sides that they don't share
have to be the same as well. So what are the sides
they don't share? Well, it's this side
and this side. So if this side is h
then this side is h. Right? But this angle is
also 60 degrees. So if we look at this 60
degrees and this 60 degrees, we know that the sides that they
don't share are also equal. Well they share this side, so
the sides that they don't share are this side and this side. So this side is h, we also
know that this side is h. Right? So it turns out that if you
have 60 degrees, 60 degrees, and 60 degrees that all the
sides have the same lengths, or it's an equilateral triangle. And that's something
to keep in mind. And that makes sense too,
because an equilateral triangle is symmetric no matter
how you look at it. So it makes sense that all of
the angles would be the same and all of the sides would
have the same length. But, hm. When we originally did this
problem we only used half of this equilateral triangle. So we know this whole side
right here is of length h. But if that whole side is of
length h, well then this side right here, just the base of
our original triangle-- and I'm trying to be messy on purpose. We tried another color. This is going to be
half of that side. Right? Because that's h over 2,
and this is also h over 2. Right over here. So if we go back to our
original triangle, and we said that this is 30 degrees and
that this is the hypotenuse, because it's opposite the right
angle, we know that the side opposite the 30 degree side
is 1/2 of the hypotenuse. And just a reminder,
how did we do that? Well we doubled the triangle. Turned it into an
equilateral triangle. Figured out this whole
side has to be the same as the hypotenuse. And this is 1/2 of
that whole side. So it's 1/2 of the hypotenuse. So let's remember that. The side opposite the 30 degree
side is 1/2 of the hypotenuse. Let me redraw that on another
page, because I think this is getting messy. So going back to what
I had originally. This is a right angle. This is the hypotenuse--
this side right here. If this is 30 degrees, we just
derived that the side opposite the 30 degrees-- it's like what
the angle is opening into-- that this is equal to
1/2 the hypotenuse. If this is equal to 1/2
the hypotenuse then what is this side equal to? Well, here we can use the
Pythagorean theorem again. We know that this side squared
plus this side squared-- let's call this side A-- is
equal to h squared. So we have 1/2 h squared plus A
squared is equal to h squared. This is equal to h squared
over 4 plus A squared, is equal to h squared. Well, we subtract h
squared from both sides. We get A squared is equal to h
squared minus h squared over 4. So this equals h squared
times 1 minus 1/4. This is equal to 3/4 h squared. And once going that's
equal to A squared. I'm running out of space,
so I'm going to go all the way over here. So take the square root of both
sides, and we get A is equal to-- the square root of 3/4
is the same thing as the square root of 3 over 2. And then the square root
of h squared is just h. And this A-- remember,
this is an area. This is what decides the
length of the side. I probably shouldn't
have used A. But this is equal to the square
root of 3 over 2, times h. So there. We've derived what all the
sides relative to the hypotenuse are of a
30-60-90 triangle. So if this is a 60 degree side. So if we know the hypotenuse
and we know this is a 30-60-90 triangle, we know the side
opposite the 30 degree side is 1/2 the hypotenuse. And we know the side opposite
the 60 degree side is the square root of 3 over 2,
times the hypotenuse. In the next module I'll show
you how using this information, which you may or may not want
to memorize-- it's probably good to memorize and practice
with, because it'll make you very fast on standardized
tests-- how we can use this information to solve the sides
of a 30-60-90 triangle very quickly. See you in the next
presentation.