The 'ratio' has become known as the golden ratio or golden
This ratio can be found in many places: in art, architecture, and mathematics.
Consider the construction of the regular
pentagon. If the side AB of a regular pentagon
(see figure to the right) has unit length,
then any diagonal, such as AC, has length
Ø = (1 + 5)/2 = 2cos(/5) = 1.61803...
and this is the golden ratio.
Indeed, in any similar pentagon/pentagram configuration the
ratio of the length of the side of the pentagram (AC) to the
length of the side of the pentagon (AB) is the golden ratio .
Notice also the diagonals of the pentagon form another regular pentagon in the center of the figure with, of course, the potential for additional diagonals to be drawn, thus generating the golden ratio again as well as another regular pentagon further inside the figure. Presumably this could continue indefinitely.
An interesting and important property of Ø is that Ø - 1 is
In this next figure (to the left) is a regular pentagon with an inscribed pentagram.|
All the line segments found there are equal in length to one of the five line
segments described below:
The length of the black line segment is 1 unit.
The length of the red line seqment, a, is Ø.
The length of the yellow line seqment, b, is 1/Ø.
The length of the green line seqment, c, is 1, like the black segment.
The length of the blue line seqment, d, is (1/Ø)2,
or equivalently, 1 - (1/Ø), as can be seen from examining the figure.
(Note that b + d = 1).
This implies that
The following ratios of the lengths of the line segments shown in the figure
are all equal to the golden ratio, Ø:
(also, red to green)
black to yellow (also, green to yellow)
yellow to blue
Ø + 1 = Ø2
Ø + 2 = ØÖ(5)
Ø3 = Ø2 + Ø
Ø3 = 2Ø + 1
Problem: Given that the perimeter of the pentagon
shown in the discussion above is 5 units, show that
the perimeter of the pentagram is
10(Ø - 1) units (6.1803... units).
Click here for a hint.
The golden ratio also appears in comparing consecutive elements of certain
kinds of sequences, most notably, the Fibonacci sequence, but other
sequences also. For instance, take two numbers
at random, say 2 and 6. Add them to get 8. Add 6 and 8 to get 14. Add
8 and 14 to get 22. If you keep this going indefinitely, then the
ratio of successive numbers approaches the golden ratio as a limit.
Hence, for starters:
Pictured here are what might be called a couple of "golden parallelograms", can
you explain why that might be?
Show that in a golden parallelogram the
length of the shorter diagonal is equal to the length
of the longer side of the parallelogram.
That is, show that the three black line
segments in the figure shown are equal.
Pictured below are some other "golden figures".
We define them as "golden" because each side can be paired with another
side such that the lengths of the pair of sides are to one another
as the golden ratio.
Also, we find the golden ratio in the following continued fraction:
The continued fraction in the white box above is just the original continued fraction,
therefore, we have the following calculations shown to the right:
Not only does Ø2 = Ø + 1, as shown, but you can show that
It's an interesting exploration to find the sequence
Golden Right Triangle I
Given that the lengths of the sides
of the triangle shown are Ö(Ø), 1 and Ø,
show that the triangle is a right triangle.
(Don't make this too hard!
And most certainly do not use a calculator.
And no, you don't know yet that the hypotenuse is
the diameter of the circle.)
Golden Right Triangle II
Given triangle ABC is inscribed in
a circle whose diameter is AB, segment CD
is perpendicular to diameter AB, and the
lengths of segments AD, BD, and CD are as shown,
find the length of segments AC and BC.
Problem: Golden Right Triangle III
Show that the triangle ABC in Problem II above is similar to the blue triangle
shown in Problem I.
Click here to visit a
'Gallery of Stars'
and Pentagrams & Pentagons.
Want to draw the Pentagram?||
Want to find the area of the Pentagram?
Problem: Given that the length of the side of the
regular pentagon shown in the discussions
above is 1 unit, express the area of the inscribed
pentagram (star) in terms of Ø.
This is a good algebra problem and
will challenge some, and there are many different
expressions that one can derive.
Here is one you can compare your result with:
If you don't get that exact expression, use a
calculator to compare your result to this.
Problem: Given that the length of the side of the
larger regular pentagon shown in the figure to the right
is 1 unit, show that the ratio of the area of larger
pentagon to the area of the smaller (white) pentagon is
Problem: And show that the ratio of the area of the
larger regular pentagon shown in the figure to the area
of the pentagram (star - red & white combined) is
For more about The Golden Ratio, see P. Davis & R. Hersh,
The Mathematical Experience, Birkhauser Pub., Chap. 4,
"Inner Issues: The Aesthetic Component".
OnLine you can follow the links below:
Pentagonal Geometry and the Golden Ratio
David Eppstein, Computer Science Department University of California, Irvine
Museum of Harmony and Golden Section
Authored by Alexey Stakhov and Anna Sluchenkova
Fibonacci Numbers and the Golden Section
This is the Home page for Ron Knott's Surrey University multimedia web site on the Fibonacci numbers, the Golden section and the Golden string.
Hosted by the Department of Mathematics of Surrey University in Guildford, UK
send comments to Thomas M. Green c/o CCC Math Dept.
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