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Special right triangles review

Learn shortcut ratios for the side lengths of two common right triangles: 45°-45°-90° and 30°-60°-90° triangles. The ratios come straight from the Pythagorean theorem.

30-60-90 triangles

30-60-90 triangles are right triangles whose acute angles are 30, degrees and 60, degrees. The sides in such triangles have special proportions:
A thirty-sixty-ninety triangle. The length of the shorter leg of the triangle is one half h units. The length of the longer leg of the triangle is square root three over two times h. The length of the hypotenuse of the triangle is h units.
Want to learn more about 30-60-90 triangles? Check out this video.

45-45-90 triangles

45-45-90 triangles are right triangles whose acute angles are both 45, degrees. This makes them isosceles triangles, and their sides have special proportions:
A forty-five-forty-five-ninety triangle. The length of both legs are k units. The length of the hypotenuse of the triangle is square root of two times k units.
The special properties of both of these special right triangles are a result of the Pythagorean theorem.
Want to learn more about 45-45-90 triangles? Check out this video.

Check your understanding

Problem 1
A right triangle A B C has angle A being thirty degrees. Side A B is x units. Side B C is six units.
A, B, equals

Want to practice more problems like this? Check out this exercise.

Want to join the conversation?

  • aqualine tree style avatar for user Brieanna Oscar
    im so used to doing a2+b2=c 2 what has changed I do not understand
    (23 votes)
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    • piceratops ultimate style avatar for user Jack Huber
      With 45-45-90 and 30-60-90 triangles you can figure out all the sides of the triangle by using only one side. If you know one short side of a 45-45-90 triangle the short side is the same length and the hypotenuse is root 2 times larger. If you know the hypotenuse of a 45-45-90 triangle the other sides are root 2 times smaller. If you know the 30-degree side of a 30-60-90 triangle the 60-degree side is root 3 times larger and the hypotenuse is twice as long. if you know the 60-degree side of a 30-60-90 triangle the 30-degree side is root 3 times smaller and the hypotenuse is 2/root 3 times longer. If you know the hypotenuse of a 30-60-90 triangle the 30-degree is half as long and the 60-degree side is root 3/2 times as long.
      (102 votes)
  • male robot hal style avatar for user Aryan
    What is the difference between congruent triangles and similar triangles?
    (11 votes)
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  • piceratops seedling style avatar for user anthony.lozano
    what can i do to not get confused with what im doing ?
    (10 votes)
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    • leaf orange style avatar for user George C
      I'd make sure I knew the basic skills for the topic. For special triangles some skills you need to master are: Angles, Square roots, and most importantly The Pythagorean Theorem. Another source you can use is the hints in the exercises, they can help guide you.
      (20 votes)
  • leafers seedling style avatar for user sydney
    How can you tell if a triangle is a 30 60 90 triangle vs a 45 45 90 triangle? Help!
    (5 votes)
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  • piceratops ultimate style avatar for user gracieseitz
    Let's say that there is a 30-60-90 triangle and I need to figure out the side opposite of the 60 degree angle and the hypotenuse is something like 6 times the square root of 3. I know that to get the answer I need to multiply this by the square root of 3 over 2. Do I multiply everything or is there a certain time when I divide or do something with square roots and/or roots? Would the answer to this problem be 36 (square root of 3 times the square root of 3 to get 3, 2 times 6 to get 12, and 12 times 3 to get 36)?
    (5 votes)
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    • winston default style avatar for user Rick
      The answer to your problem is actually 9. You are correct about multiplying the square root of 3 / 2 by the hypotenuse (6 * root of 3), but your answer is incorrect. This is because if you multiply the square root of 3 by 6 times the root of three, that would be the same as multiplying 3 by 6 (because the square root of 3 squared is 3). 3 by 6 is 18, and that divided by 2 would equal 9, which is the correct answeer.
      (8 votes)
  • blobby green style avatar for user Esa Abuzar
    if I get 30.1 degrees, is it still a special triangle
    (5 votes)
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  • duskpin tree style avatar for user Siena
    Can't you just use SOH CAH TOA to find al of these?
    (5 votes)
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  • duskpin ultimate style avatar for user april_oh_
    I use this trick on 30, 60, 90 triangles and I've never gotten a single wrong -
    1. The small leg to the hypotenuse is times 2, Hypotenuse to the small leg is divided by 2.
    2. The small leg (x) to the longer leg is x radical three

    For Example-
    Pretend that the short leg is 4 and we will represent that as "x." And we are trying to find the length of the hypotenuse side and the long side. To find the lengths of the hypotenuse from the short leg (x), all we have to do is x times 2, which in this case is 4 times 2. Four times 2 is 8. The length of the hypotenuse side is 8. That is how to find the hypotenuse from the short leg. But are we done yet? No, we are not. We still have to find the length of the long leg. Since the short leg (x) is 4, we have to do "x" radical three. I came to a conclusion that the long leg is 4 radical 3.

    -This works everytime
    (5 votes)
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  • male robot hal style avatar for user jinseo.park
    Are special right triangles still classified as right triangles?
    (2 votes)
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    • hopper jumping style avatar for user Markarino /TEE/DGPE-PI1   #Evaluate
      Boy, I hope you're still around. I hate that nobody has answered this very good question.

      The short answer is, yes.

      Unfortunately, I'm new around here, but I can tell you what I understand. I don't know if special triangles are an actual thing, or just a category KA came up with to describe this lesson. What I can tell you is that the special triangles that they describe here in these lessons are the 30-60-90 triangle, which is always a right triangle (because of the 90 degree angle) and the 45-45-90 right triangle.
      (8 votes)
  • blobby green style avatar for user Francesco Blz
    In a triangle 30-60-90, if I am given the long side as an integer, how can I derive the calculation of the other sides?
    So, for instance, if I have 18 as the side that corresponds to the ratio square root of 3, how do I manage the proportions to figure out the other sides (hypothenuse or short side)?
    Thanks!
    (2 votes)
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    • mr pink green style avatar for user David Severin
      If you start with x√3 = 18, divide both sides by √3 to get x = 18/√3, but since we do not like roots in the denominator, we then multiply by √3/√3 to get 18√3/(√3*√3) = 18 √3/3=6√3. Doubling to get the hypotenuse gives 12√3.
      So it does not matter what the value is, just multiply this by √3/3 to get the short side. That pattern works for 45-45-90 with x-x-x√2. If the hypotenuse is a number like 18, multiply it by √2/2 to get the sides to be 9√2.
      (5 votes)