Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
K12 LibreTexts

4.43: 30-60-90 Right Triangles

( \newcommand{\kernel}{\mathrm{null}\,}\)

Hypotenuse equals twice the smallest leg, while the larger leg is 3 times the smallest.

One of the two special right triangles is called a 30-60-90 triangle, after its three angles.

30-60-90 Theorem: If a triangle has angle measures 30, 60 and 90, then the sides are in the ratio x:x3:2x.

The shorter leg is always x, the longer leg is always x3, and the hypotenuse is always 2x. If you ever forget these theorems, you can still use the Pythagorean Theorem.

What if you were given a 30-60-90 right triangle and the length of one of its side? How could you figure out the lengths of its other sides?

Example 4.43.1

Find the value of x and y.

f-d_f253e0915a3a06ec0d626be4e6d23bcd759bdf5480908ad2fcd50348+IMAGE_TINY+IMAGE_TINY.png
Figure 4.43.1

Solution

We are given the longer leg.

x3=12x=12333=1233=43The hypotenuse isy=2(43)=83

Example 4.43.2

Find the value of x and y.

f-d_54966f54aaec43bfd1b984a46d9b523714d601e30028766637c91a37+IMAGE_TINY+IMAGE_TINY.png
Figure 4.43.2

Solution

We are given the hypotenuse.

2x=16x=8The longer leg isy=83=83

Example 4.43.3

Find the length of the missing sides.

f-d_6cf56285ac6d4e253c51867d19ce5f1cf4dd5eb5ccd2c207bc43f65d+IMAGE_TINY+IMAGE_TINY.png
Figure 4.43.3

Solution

We are given the shorter leg. If x=5, then the longer leg, b=53, and the hypotenuse, c=2(5)=10.

Example 4.43.4

Find the length of the missing sides.

f-d_382c5686a01a4042d0dc31e15fa8fa9f0185f564b927cd65b93dc128+IMAGE_TINY+IMAGE_TINY.png
Figure 4.43.4

Solution

We are given the hypotenuse. 2x=20, so the shorter leg, f=202=10, and the longer leg, g=103.

Example 4.43.5

A rectangle has sides 4 and 43. What is the length of the diagonal?

f-d_bfba1fae7bcbddafcff19d9ed2213028dcb883213bc48f78ea211ab7+IMAGE_TINY+IMAGE_TINY.png
Figure 4.43.5

Solution

The two lengths are x, x3, so the diagonal would be 2x, or 2(4)=8.

If you did not recognize this is a 30-60-90 triangle, you can use the Pythagorean Theorem too.

42+(43)2=d216+48=d2d=64=8

Review

  1. In a 30-60-90 triangle, if the shorter leg is 5, then the longer leg is __________ and the hypotenuse is ___________.
  2. In a 30-60-90 triangle, if the shorter leg is x, then the longer leg is __________ and the hypotenuse is ___________.
  3. A rectangle has sides of length 6 and 63. What is the length of the diagonal?
  4. Two (opposite) sides of a rectangle are 10 and the diagonal is 20. What is the length of the other two sides?

For questions 5-12, find the lengths of the missing sides. Simplify all radicals.

  1. f-d_2facd36bb2fbeedd849497dd107ba534533e6a88784cf8456c710384+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.43.6
  2. f-d_e677a666e07da6787e00079bd4c8b21ab821694093fbfc62681e204e+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.43.7
  3. f-d_1f29f072cb73160e7892970886b0c1fcebf8125b24b8ba11683654ae+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.43.8
  4. f-d_c4d28ed45dc897a6ead6481b0de495a862a1228b6a361f6912b49eef+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.43.9
  5. f-d_39db2d0dd4c8617180356f22daf33f5009d5e91ecfc8c679fd16c0fb+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.43.10
  6. f-d_e757021a70a51b662a00b9af57195087812eade33c7db373cf0705d9+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.43.11
  7. f-d_9c3a73a85f62650dafe5157f8ee7dd59ccc48a4a6631f25d1728a0fc+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.43.12
  8. f-d_6a40679372417fa67db43201a11e0a3f5cb60ec698a15303644d8570+IMAGE_TINY+IMAGE_TINY.png
    Figure 4.43.13

Review (Answers)

To see the Review answers, open this PDF file and look for section 8.6.

Resources

Vocabulary

Term Definition
30-60-90 Theorem If a triangle has angle measures of 30, 60, and 90 degrees, then the sides are in the ratio x:x3:2x
30-60-90 Triangle A 30-60-90 triangle is a special right triangle with angles of 30, 60, and 90.
Hypotenuse The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle.
Legs of a Right Triangle The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle.
Pythagorean Theorem The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by a2+b2=c2, where a and b are legs of the triangle and c is the hypotenuse of the triangle.
Radical The \boldsymbol{\sqrt}, or square root, sign.

Additional Resources

Interactive Element

Video: Solving Special Right Triangles

Activities: 30-60-90 Right Triangles Discussion Questions

Study Aids: Special Right Triangles Study Guide

Practice: 30-60-90 Right Triangles

Real World: Fighting the War on Drugs Using Geometry and Special Triangles


This page titled 4.43: 30-60-90 Right Triangles is shared under a CK-12 license and was authored, remixed, and/or curated by CK-12 Foundation via source content that was edited to the style and standards of the LibreTexts platform.

CK-12 Foundation
LICENSED UNDER
CK-12 Foundation is licensed under CK-12 Curriculum Materials License

Support Center

How can we help?