30-60-90 Triangle | Theorem, Ratio, & Formula (video)

What is a 30-60-90 Triangle?

A 30-60-90 triangle is a right triangle where the three home angles measure 30°, 60°, and 90° .
What Is A 30-60-90 Triangle
right triangles with 30-60-90 department of the interior angles are known as special right triangles. special triangles in geometry because of the potent relationships that unfold when studying their angles and sides .
In all triangles, the relationships between angles and their antonym sides are easy to understand. The greater the angle, the longer the opposition side.

Triangle Relationship Between Angles and Opposite Sides
This means, of the three interior angles, the largest home angle is reverse the longest of the three sides, and the smallest angle will be opposite the shortest slope .
In a mighty triangle, recall that the slope opposite the correct angle ( the largest slant ) is called the hypotenuse ( the longest side, and the other two sides are called legs .

Table Of Contents

30-60-90 Triangle Ratio

A 30-60-90 degree triangulum is a special right triangulum, so it ‘s side lengths are always consistent with each other. The ratio of the sides follow the 30-60-90 triangulum ratio :

1 : 2 : 3

  • short side ( opposite the 30 academic degree lean ) = adam
  • Hypotenuse ( opposite the 90 degree angle ) = 2x
  • long side ( opposite the 60 degree slant ) = x3

30-60-90 Triangle Ratio And Theorem

30-60-90 Triangle Theorem

These three special properties can be considered the 30-60-90 triangle theorem and are unique to these particular right triangles :

  1. The hypotenuse ( the triangle ‘s longest side ) is always twice the duration of the short leg
  2. The duration of the longer leg is the inadequate leg ‘s duration times 3
  3. If you know the length of any one side of a 30-60-90 triangle, you can find the miss side lengths
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other interesting properties of 30-60-90 triangles are :

  • All 30-60-90 triangles are exchangeable
  • Two 30-60-90 triangles sharing a long peg shape an equilateral triangle

Other Interesting Properties Of Special Right Triangles

How to Solve a 30-60-90 Triangle

education is knowing that 30-60-90 triangles have three properties laid out in the theorem. Wisdom is knowing what to do with that cognition. Suppose you have a 30-60-90 triangle :
How To Solve A 30-60-90 Triangle
We know that the hypotenuse of this triangulum is doubly the length of the light leg :
3.46 km2 = 1.73 kilometer
We besides know that the long stage is the brusque stage multiplied times the square settle of 3 :
1.73 × 3 ≈ 3 kilometer
We set up our special 30-60-90 to showcase the simplicity of finding the distance of the three sides. Try figuring this one forbidden :
30-60-90 Triangle Example Problem
The retentive branch is the short leg times 3, so can you calculate the short-circuit leg ‘s length ? Did you say 5 ?
The length of the hypotenuse is always doubly the short stage ‘s length. Did you get 10 ?
You can create your own 30-60-90 Triangle convention using the known information in your problem and the follow rules. This table of 30-60-90 triangle rules to help you find miss side lengths :

30-60-90 Triangle Rules

If you know… Then… To get…
Hypotenuse Divide by 2

Short leg
Short leg Multiply by 2 Hypotenuse
Short leg Multiply by 3 Long leg
Long leg Divide by 3 Short leg
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When working with 30-60-90 triangles, you may be tempted to force a relationship between the hypotenuse and the long leg. That relationship is challenging because of the square root of 3 .
Work cautiously, concentrating on the relationship between the hypotenuse and short stage, then short branch and long leg .
You will notice our examples so far only provide information that would “ plugin ” easily using our three properties. sometimes the geometry is not so easy .
30-60-90 Triangle Rules
What if the retentive stage is labeled with a simple, wholly number ?
You leap into the problem since getting the short leg is simply a matter of dividing the retentive leg by the hearty root of 3, then doubling that to get the hypotenuse .
But you can not leave the trouble like this :
27 cm3
The rules of mathematics do not permit a radical in the denominator, so you must rationalize the fraction. Multiply both numerator and denominator times 3 :
27 cm3 × 33
This simplifies to :
27 39 = 27 33 = 9 3
Unless your directions are to provide a decimal answer, this can be your concluding answer for the duration of the curtly side. Doubling this gives 18 3 for the hypotenuse .
Another warn flag with 30-60-90 triangles is that you can become thus steep in the three properties that you lose sight of the triangulum itself. It is still a triangle, so its department of the interior angles must add to 180°, and its three sides must still adhere to the Pythagorean Theorem :


a2 + b2 = c2

You can use the Pythagorean Theorem to check your work or to jump-start a solution .
30-60-90 Triangle And Pythagorean Theorem

30-60-90 Triangle Examples

A right triangle has a short side with a duration of 14 meters with the opposition lean measuring 30°. What are the other two lengths ?
30-60-90 Triangle Finding Missing Lengths
We know immediately that the triangulum is a 30-60-90, since the two identified angles sum to 120° :
180° – 120° = 60°
The missing slant measures 60°. It follows that the hypotenuse is 28 m, and the long leg is 14 molarity * 3 .
What is you have a triangulum with the hypotenuse labeled 2,020 millimeter, the short-circuit leg labeled 1,010 millimeter, and the long leg labeled 1,0103 .
Your cognition of the 30-60-90 triangle will help you recognize this immediately. You can confidently label the three inside angles because you see the relationships between the hypotenuse and short leg and the light leg and hanker leg .
here is a more detailed problem. What is the length of the inadequate branch, line segment MH ? :

Special Triangle Example Problem
Did you say 50 inches ? This is truly two 30-60-90 triangles, which means hypotenuse MA is besides 100 inches, which means the shortest leg MH is 50 inches .

Next Lesson:

Triangle Congruence Theorem

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