Constructions for the Golden Ratio
Here is a similar problem but this time using circles: to remove the largest circle and still have a pivotal point:
The full circle balances at its centre point 
Removing a small circle at the top moves the balance point down a little 
Removing a large circle moves the balance point outside the crescent shape 
The limit point is when the radii of the two circles are in the golden ratio and the centre of gravity is just on the edge of the inner circle 
Ellipse and Ring
The third little geometrical gem is about a ring and an ellipse. An ellipse is the shape that a plate or anything circular appears when viewed at an angle.
Here are two circles, one inside the other. The yellow and red areas define an outer ring. Also the orange and red parts form an ellipse (an oval).  If the ring is very narrow, the two circles are similar in size and the ellipse has a much bigger area than the ring.  If the inner circle is very small, the ellipse will be very narrow and the outer ring will be much bigger in area than the ellipse. 
So the question is
The answer is again when the inner radius is 0.618 of the larger one, the golden ratio.
This is quite easy to prove using these two formulae:
The are of a circle of radius r is π r^{2}
The area of an ellipse with "radii" a and b (as shown above) is πab
(Note how that, when a = b in the ellipse, it
becomes a circle and the two formulae are the same.)
So the outer circle has radius a, the inner
circle radius b and the area of the ring
between them
is therefore:
Area of ring = π (b^{2} – a^{2})
This is equal to the area of the ellipse when
π (b^{2} – a^{2}) = π a b
b^{2}  a^{2}  a b = 0
If we let the ratio of the two circles radii = b/a, be K, say, then dividing the equation by a^{2} we have
K^{2}  K  1=0
which means K is either Phi or –phi. The positive value for K means that b = Phi a or a = phi b.
The equation of an ellipse is
(x/b)^{2} + (y/a)^{2} = 1
When a = b, we have the equation of a circle of radius a(=b)
:
(x/a)^{2} + (y/a)^{2} = 1 or
x^{2} + y^{2} = a^{2} as it is more usually written.
You might have spotted that this equation is merely Pythagoras' Theorem that all the points (x,y) on the circle are the same distance from the origin, that distance being a.
Note 79.13 A Note on the Golden Ratio, A D Rawlings,
Mathematical
Gazette vol 79 (1995) page 104.
The
Changing Shape of Geometry C Pritchard (2003) Cambridge University
Press paperback
and hardback,
is a collection of popular, interesting and enjoyable articles selected
from the
Mathematical
Gazette . It will be of particular interest to teachers and
students in school
or indeed anyone interested in Geometry.
The three gems above are given in more
detail in the section on The Golden Ratio.
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The Fibonacci Spiral and the Golden Spiral
The Fibonacci Squares Spiral
On the Fibonacci Numbers and Golden Section in Nature page, we looked at a spiral formed from squares whose sides had Fibonacci numbers as their lengths.
This section answers the question:
The Golden section squares are shown in red here, the axes in blue
and all the points of the
squares lie on the green lines, which pass through the origin (0,0).
Also, the blue (axes) lines and the green lines are each separated
from the next
by 45° exactly.
The large rectangle ABDF is the exactly the same shape as
CDFH,
but is (exactly) phi times as large. Also
it has been rotated by a quarter turn. The same applies to CDFH and
HJEF
and to all the golden rectangles in the diagram.
So to transform OE (on the x axis) to OC
(on the y axis), and indeed any point on the
spiral
to another point on the spiral, we expand lengths by phi times for every rotation of 90°:
that is, we change (r,theta) to (r Phi,theta + π/2) (where, as usual,
we express angles in radian measure, not degrees).
So if we say E is at (1,0),
then C is at (Phi,π/2),
A is at (Phi^{2}, π), and so on.
Similarly, G is at (phi,–π/2),
and I is at (phi^{2}, –π) and so on
because phi = 1/Phi.
The points on the spiral are therefore summarised by:
If we eliminate the n in the two equations, we get a single equation that all the points on the spiral satisfy:
or
For ordinary (cartesian) coordinates, the x values are y values are
generated from the polar coordinates as follows:
x = r cos(theta)
y = r sin(theta)
which we can then use in a Spreadsheet to generate a chart as shown
here.
Such spirals, where the distance from the origin is a constant to the
power of the angle,
are called equiangular spirals. They also have the property
that
a line from the origin to any point
on the curve always finds (the tangent to) the curve meeting it at
the same angle.
Another name is a logarithmic spiral because the angle of any
point
from the x axis through the origin
is proportional to the logarithm of the point's distance from the
origin.
To see that the Fibonacci Spiral here is only an approximation to the (true) Golden Spiral above note that:

The Golden or Phi Spiral
In the spiral above, based on Fibonacci squares spiralling out from an
initial two 1x1 squares,
we noted that one quarter turn produces an expansion by Phi in the distance of a point on the curve
from the "origin". So in one full turn we have an expansion of Phi^{4}.
In seashells, we notice an expansion of Phi in one turn, so that not only has the
shell grown to Phi times as far from its
origin (now buried deep inside the shell).
Also, because of the properties of the golden section, we can see that
the distances measured
on the outside of the shell also have increased by Phi and it is often easier to measure this distance
on the outside of a shell, as we see in the picture here on the left.
In this case, the equation of the curve is
r = Phi ^{ theta / 360 } if theta is in degrees.
Notice that an increase in the angle theta of
2 π radians (360° or
one full turn)
makes r increase by a factor of Phi because the power of Phi has increased by 1.
I will call this spiral, that increases by Phi per turn, the Golden Spiral or the Phi Spiral because of this property and also because it
is the one we find in
nature (shells, etc.).
Click on the Spreadsheet image to open an Excel Spreadsheet to generate
the Golden Spiral
in a new window.
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Trigonometry and Phi
What is trigonometry?
We can answer this by looking at the origin of the word trigonometry.
Words ending with metry are to do with measuring (from
the Greek
word metron meaning "measurement").
(What do you think that thermometry measures?
What about geometry? Can you think of any more words ending
with metry?)
Also, the gon part comes from the Greek gonia) meaning angle.
It is derived from the Greek word for "knee" which is gony.
The prefix tri is to do with three as in tricycle (a threewheeled cycle), trio (three people), trident
(a threepronged fork).
Similarly, quad means 4, pent 5 and
hex six as in the following:
 a (fivesided and) fiveangled shape is a pentagon meaning literally fiveangles and
 a six angled one is called a hexagon then we could
call
 a fourangled shape a quadragon
(but we don't  using the word quadrilateral instead which means "foursided") and  a threeangled shape would be a triagon
(but we call it a triangle instead)
"Trigon" was indeed the old English word for a triangle.
So trigonon means
"threeangled" or, as we would now say in English, "triangular" and
hence we have
trigoniametria meaning "the measurement of triangles".
With thanks to proteus of softnet for this information.