Constructions for the Golden Ratio

Phi and Pentagons

There is an intimate connection between the golden section and the regular 5-sided shape called a pentagon and its variation - the pentagram - that we explore first.

Pentagons and Pentagrams

Pentagram The pentagram is a symmetrical 5-pointed star that fits inside a pentagon. Starting from a pentagon, by joining each vertex to the next-but-one you can draw a pentagram without taking your pen off the paper.

The pentagram has 5 triangles on the edges of another pentagon at its centre. Let's focus on one of the triangles and the central pentagon as shown here.

pent and angles All the orange angles at the vertices of the pentagon are equal. They are called the external angles of the polygon. What size are they? This practical demonstration will give us the answer:

  • Take a pen and lay it along the top edge pointing right
  • Turn it about the top right vertex through the orange angle so that it points down to the lower right
  • Move the pen down that side of the pentagon to the next vertex and turn it through the next orange angle
  • Repeat moving it along the sides and turning through the rest of the orange angles until it lies back on the top edge
  • The pen is now back in its starting position, pointing to the right so it has turned through one compete turn.
  • It has also turned through each of the 5 orange angles
  • So the sum of the 5 orange angles is one turn or 360°
  • Each orange angle is therefore 360/5=72°

The green angle is the same size as the orange angle so that the two "base" angles of the blue triangle are both 72°.
Since the angles in a triangle sum to 180° the yellow angle is 36° so that 72° + 72° + 36° = 180°.

The basic geometrical facts we have used here are:

The external angles in any polygon sum to 360°.
The angles on a straight line sum to 180°.
The angles in a triangle sum to 180°.

So the pentagram triangle has angles of 36 °, 72° and 72°. Now let's find out how long its sides are.

The 36°-72°-72° triangle

Isoceles 36-36-72 angle triangle In this diagram, the triangle ABC is isosceles, since the two sides, AB and AC, are equal as are the two angles at B and C.
[Also, angles ABC and ACB are twice angle CAB.]

If we bisect the base angle at B by a line from B to point D on AC then we have the angles as shown and also angle BDC is also 72°. BCD now has two angles equal and is therefore an isosceles triangle; and also we have BC=BD.

Since ABD also has two equal angles of 36°, it too is isosceles and so BD=AD. So in the diagram the three sides BC, BD and DA are all the same length.

We also note that the little triangle BCD and the whole triangle ABC are similar since they are both 36°-72°-72° triangles.

Let's call the smallest segment here, CD, length 1 and find the lengths of the others in relation to it. We will therefore let the ratio of the smaller to longer sides in triangle BCD be r so that if CD is 1 then BC is r.

In the larger triangle ABC, the base is now r and as it is the same shape as BCD, then its sides are in the same ratio so Ab is r times BC, e.d. AB is r2.
Also, we have shown BC=BD=AD so AD is r (and CD is 1).
From the diagram we can see that AC=AD+DB.
But AC=1+r and in isosceles triangle ABC, AB (which is r2) is the same length as AC (which is 1+r), so
r2 = 1 + r
and this is the equation which defined the golden ratio.

r is Phi or -phi and since lengths are positive, we therefore have that r is Phi!

So the triangle with angles 36°, 72°, 72° has sides that are proportional to Phi, Phi and 1 (which is the same as 1,1,phi). 72,72,36 degree triangle has sides P,P and 1

Pentagrams and the 36°-72°-72° triangle

If we look at the way a pentagram is constructed, we can see there are lots of lines divided in the golden ratio: Since the points can be joined to make a pentagon, the golden ratio appears in the pentagon also and the relationship between its sides and the diagonals (joining two non-adjacent points).

The reason is that Phi has the value 2 cos (π/5) where the angle is described in radians, or, in degrees, Phi=2 cos (36°).
[See below for more angles whose sines and cosines involve Phi!]

Pentagram with fib lengths

Since the ratio of a pair of consecutive Fibonacci numbers is roughly equal to the golden section, we can get an approximate pentagon and pentagram using the Fibonacci numbers as lengths of lines:
There is another flatter triangle inside the pentagon here. Has this any golden sections in it? Yes! We see where further down this page, but first, a quick and easy way to make a pentagram without measuring angles or using compasses: