The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Golden Triangle: What is It?

Date: 09/19/1999 at 16:27:01
From: Robin Byerly
Subject: Golden Triangle

I'm doing a project for math where I have to answer questions using 
the Internet. I have gone to many different Web sites trying to figure 
out what the Golden Triangle is, but I cannot find it. It has to do 
with math but so far all my results have something to do with 
bicycles. Please help me find out what the Golden Triangle is. Thank 

Robin Byerly

Date: 09/20/1999 at 05:34:49
From: Doctor Floor
Subject: Re: Golden Triangle

Hi, Robin,

Thanks for your question.

Suppose we have a triangle ABC, such that <A = <B = 72 degrees 
(< means angle) and <C = 36 degrees. Such a triangle is known as the 
Golden Triangle.


Let D be the point on BC, such that AD is the angle bisector of <A. 
Then triangle ABD is again a Golden Triangle.


When we let lengths AB = AD = CD = x and BD = 1, then we find:

     AB : BD  =  BC : AB
      x : 1   = (x+1) : x

This can be rewritten to:

            x^2 = x + 1
     x^2 -x - 1 = 0

The two solutions for x are x = 1/2 +/- sqrt(5)/2. Since AB > BD, in 
this case we must have x = 1/2 + sqrt(5)/2.

The number 1/2 + sqrt(5)/2 is known as the Golden Ratio, or Golden 
Mean. So BC : AB is this famous ratio; that's why this triangle is 
called a Golden Triangle.

For more about the Golden Ratio, see our FAQ:

  Golden Ratio, Fibonacci Sequence   

As an example of the appearance of Golden Triangles: the outside 
triangles of a pentagram are Golden Triangles.

When we attach to AC and BC two triangles that are congruent to 
triangle ACD, we find a regular pentagon.


I hope this helped. If you need more, just write us back.

Best regards,
- Doctor Floor, The Math Forum   
Associated Topics:
High School Definitions
High School Fibonacci Sequence/Golden Ratio
High School Geometry
High School Triangles and Other Polygons
Middle School Definitions
Middle School Geometry
Middle School Triangles and Other Polygons

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.