The Golden Ratio

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The Golden Ratio
Begin by drawing a square with sides of 10cm
10 cm
10 cm
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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
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The Golden Ratio
Notice that the square has been divided into two
equal rectangles
10 cm
x
10 cm
x
5 cm
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The Golden Ratio
Now draw a diagonal in the 2nd rectangle as shown
10 cm
x
10 cm
x
5 cm
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The Golden Ratio
Set your compass to the length of this diagonal
10 cm
x
10 cm
x
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The Golden Ratio
Use the compass to draw an arc as shown
10 cm
x
10 cm
And extend the bottom line of the square to meet
the arc
x
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The Golden Ratio
Carefully measure the line AB
10 cm
x
Write down yourmeasurement in centimetres to 1
decimalplace
10 cm
x
A
B
? cm
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The Golden Ratio
Now complete the rectangle
Write down the ratio of long side short side
10 cm
x
??.? 10
10 cm
Express this as a fraction
??.? 10
x
A
B
? cm
And convert to a decimal
??.? 10
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The Golden Ratio
Next we will perform an EXACT calculation for the
ratio of long side short side
E
10 cm
Consider the diagonal DE
Use Pythagoras in triangle DCE to find DE (try
it now)
10 cm
a² b² c²
DE² DC² CE²
A
B
DE² 5² 10²
D
C
5
Notice that DB DE
So AB AD DB
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The Golden Ratio
Next we will perform an EXACT calculation for the
ratio of long side short side
E
10 cm
So the ratio is
long side short side
AB AC
10 cm
10
And as a fraction canbe expressed as
A
B
D
C
5
and as a decimal is
1.61803
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The Golden Ratio
Now look closely at the smaller rectangle which
has formed inside the large one as shown
The long side, CE is obviously 10 cm
Can you determine an expression for the short
side CB?
CB DB DC
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The Golden Ratio
We can now find the ratio of long side short
side for this small rectangle
long side short side
CE CB
Convert the ratio into a fraction and then to a
decimal as before (try it now)
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The Golden Ratio
We can now find the ratio of long side short
side for this small rectangle
long side short side
CE CB
As a fraction is
And as a decimal is
1.61803
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The Golden Ratio
So BOTH rectangles.. The BIG one
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The Golden Ratio
So BOTH rectangles..
and the SMALL one
have
long side short side
In the SAME ratio of
1.61803
And they share a common side
This ratio is very special and is given the
symbol F (Phi) It is commonly referred to as The
Golden Ratio
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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
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)
x
1
So ABAC ACCB
Write this relationship in fraction fromthen
substitute the lengths in. (try it now)
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The Golden Ratio
You should now have the following working.
By coss-multiplying, find a quadratic equation to
solve.(try it now)
You should end up with the equation
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The Golden Ratio
This quadratic cannot be readily factorised so
try using the quadratic formula to solve it. (try
it now)
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The Golden Ratio
Find the two solutions in decimal form to 5
decimal places. Any comments about the two
answers?
So x 1.61803 or x -0.61803
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!
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The Golden Ratio
More to explore
http//www.mathsisfun.com/numbers/golden-ratio.htm
lhttp//www.bbc.co.uk/learningzone/clips/the-beau
ty-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