Home
Arithmetic
Algebra
Geometry
Word Problems
Statistics
Probability
Set Theory
Trigonometry
Matrices
SAT Preparation
ACT Preparation
GMAT Preparation
Math Worksheets
Math Games
Math Trivia
Chemistry
Animal Facts
Links
 

30-60-90 Right Triangles

Recognizing special right triangles in geometry can help you to answer some questions quicker. A special right triangle is a right triangle whose sides are in a particular ratio. You can also use the Pythagorean theorem, but if you can see that it is a special triangle it can save you some calculations.

 

 

There are three types of special triangles: 3-4-5 triangles, 45°-45°-90° triangles and 30°-60°-90° triangles. In this lesson we will study 30°-60°-90° triangles.

 

30º-60º-90º Triangles

Another type of special right triangles is the 30°- 60°- 90° triangle. This is right triangle whose angles are 30°, 60°and 90°. The lengths of the sides of a 30°- 60°- 90° triangle are in the ratio of 1:root3:2 .

             n:n:root3:2

30-60-90 rt triangle

Example 1:

Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are 4 inches and 4 root 3  inches.

Solution:

Step 1: Test the ratio of the lengths to see if it fits the n:n: root3:2  ratio.             

  n:n root3:2

Step 2:  Yes, it is a 30°- 60°- 90° triangle for n = 4

Step 3:  Calculate the third side.

2n = 2×4 = 8

Answer: The length of the hypotenuse is 8 inches.

You can also recognize a 30°- 60°- 90° triangle by the angles. As long as you know that one of the angles in the right-angle triangle is either 30° or 60° then it must be a 30°- 60°- 90° special right triangle.

A right triangle with a 30° angle or 60° angle must be a 30°- 60°- 90° special right triangle.

 

 

Example 2:

Find the lengths of the other two sides of a right triangle if the length of the hypotenuse is 8 inches and one of the angles is 30°.

Solution:

Step 1: This is a right triangle with a 30° angle so it must be a 30°- 60°- 90° triangle.

You are given that the hypotenuse is 8. Substituting 8 into the third value of the ratio n:n:root3:2 , we get that 2n = 8 Þ n = 4.

Substituting n = 4 into the first and second value of the ratio we get that the other two sides are 4 and 4 root 3 .

Answer: The lengths of the two sides are 4 inches and 4 root 3 inches.

 

 

The following videos give more examples of 30-60-90 triangles.

 

 

 

Custom Search

 

We welcome your feedback, comments and questions about this site - please submit your feedback via our Feedback page.

 

© Copyright 2005, 2008 - onlinemathlearning.com
Embedded content, if any, are copyrights of their respective owners.

Useful Links:
More Geometry Help on MathWorld

 

 

Custom Search