The Golden Ratio
Begin by drawing a square with sides of 10cm
10 cm
10 cm
The Golden Ratio
Make a 5 cm mark on the top and bottom of the
square and connect with a line
10 cm
x
10 cm
x
5 cm
The Golden Ratio
Notice that the square has been divided into two
equal rectangles
10 cm
x
10 cm
x
5 cm
The Golden Ratio
Now draw a diagonal in the 2nd rectangle as shown
10 cm
x
10 cm
x
5 cm
The Golden Ratio
Set your compass to the length of this diagonal
10 cm
x
10 cm
x
The Golden Ratio
Use the compass to draw an arc as shown
10 cm
x
10 cm
x
And extend the bottom line
of the square to meet the arc
The Golden Ratio
Carefully measure the line AB
10 cm
x
Write down your
measurement in
centimetres to 1 decimal
place
10 cm
A
x
B
? cm
The Golden Ratio
Now complete the rectangle
Write down the ratio of
long side : short side
10 cm
x
??.? : 10
10 cm
A
x
B
Express this as a fraction
??.?
10
? cm
And convert to a decimal
??.? ÷ 10 =
The Golden Ratio
Next we will perform an EXACT calculation for the
ratio of …… long side : short side
10 cm
E
Consider the diagonal DE
Use Pythagoras in triangle
DCE to find DE (try it now)
5 5
10 cm
A
D
5
C
B
5 5
Notice that DB = DE
So AB = AD + DB = 5  5 5
a² = b² + c²
DE² = DC² + CE²
DE² = 5² + 10²
DE 
125
DE 
25  5
DE  5 5
The Golden Ratio
Next we will perform an EXACT calculation for the
ratio of …… long side : short side
10 cm
E
So the ratio is…
long side : short side
5 5
AB : AC
10 cm
5  5 5 : 10
A
D
5
C
5 5
B
And as a fraction can
be expressed as
55 5
10
and as a decimal is 1.61803
The Golden Ratio
Now look closely at the smaller rectangle which has
formed inside the large one as shown…
10 cm
The long side, CE
is obviously 10 cm
E
5 5
Can you determine an
expression for the
short side CB?
10 cm
A
D
5
C 5 5 5
5 5
B
CB = DB – DC
CB  5 5  5
The Golden Ratio
We can now find the ratio of long side : short side
for this small rectangle…
10 cm
long side : short side
E
CE : CB
10 : 5 5  5
10 cm
A
D
5
C 5 5 5
5 5
B
Convert the ratio into a
fraction and then to a decimal
as before (try it now)
The Golden Ratio
We can now find the ratio of long side : short side
for this small rectangle…
10 cm
long side : short side
E
CE : CB
10 : 5 5  5
10 cm
As a fraction is …
A
D
5
C 5 5 5
10
B
5 5 5
5 5
And as a decimal is
10  ( 5 5  5 )
= 1.61803
The Golden Ratio
So BOTH rectangles….. The BIG one
10 cm
E
10 cm
A
D
5
C 5 5 5
5 5
B
The Golden Ratio
So BOTH rectangles…..
10 cm
and the SMALL one
have
E
long side : short side
10 cm
In the SAME ratio of
1.61803
A
D
5
C 5 5 5
B
And they share a common side
5 5
This ratio is very special and is given the symbol Φ (Phi)
It is commonly referred to as The Golden Ratio
The Golden Ratio
Here is another, faster way to find the Golden Ratio.
Consider the same rectangle, drawn the same way.
Let the distance AC = x
And the distance CB = 1
A
B
C
x
1
Remember that the sides
of the big rectangle must be
in the SAME ratio as the sides
of the small rectangle
(they are SIMILAR rectangles)
So AB:AC = AC:CB Write this relationship in fraction from
then substitute the lengths in.
(try it now)
The Golden Ratio
You should now have the following working….
AB
AC
CB
x 1
x
x
A
B
C
x
1

AC

1
By coss-multiplying, find a
quadratic equation to solve.
(try it now)
You should end up with the equation
x  x 1  0
2
The Golden Ratio
This quadratic cannot be readily factorised so try
using the quadratic formula to solve it. (try it now)
x  x 1  0
2
a  1, b   1 and c   1
x
A
x
1
b  4 ac
2
2a
B
C
b
x
1
1  4 (1)(  1)
2
2
x
1
5
2
The Golden Ratio
Find the two solutions in decimal form to 5 decimal
places. Any comments about the two answers?
x  x 1  0
2
So x = 1.61803
or x = -0.61803
A
B
C
x
1
We can discount the negative
answer as it makes little sense
in terms of a length.
The negative answer does however contain exactly the same
digits after the point as the positive answer which is a bit strange!
The Golden Ratio
More to explore
http://www.mathsisfun.com/numbers/golden-ratio.html
http://www.bbc.co.uk/learningzone/clips/the-beauty-of-the-golden-ratio/9017.html
Also, the negative answer is interesting and worth finding out
more about.
Fibonacci numbers are directly related to the
Golden Ratio – find out why.
K.Rybarczyk 2011