The Regular Pentagon
A regular pentagon is a pentagon in which all sides are equal and all of the angles at the vertices are equal. In a regular polygon, the angle at each vertex is given by the formula:
So, for a regular pentagon n = 5 and we have
Figure 5.1. shows a regular pentagon with all of the vertices and edges labeled.
The triangle ACD is formed by drawing lines between non-adjacent vertices.
This forms three isosceles triangles i.e. a triangle with two equal sides: ABC, AED and ACD. In the two outer triangles, ABC AED, the angle at vertices B and E is 108 degrees and hence the angle at the other two vertices is 36 degrees (180 108)/2. In the triangle ACD, the angle at A is 36 degrees (108 36 36)
and the angles at C and D are 72 degrees (108 36). In triangle ACD, the angle at vertex C has been bisected forming the two additional triangles ACF and CDF.
The angle in the triangle CDF at vertex F is 72 degrees ( 180 72 36 ) and the angle in triangle ACF at vertex F is 108 degrees ( 180 72 ).
Since triangles ACD and CDF have identical angles, the triangles are said to be similar. This means that the ratios of corresponding sides are also equal. In triangle CDF side CF is 1 and in triangle ACF side AF is also 1. This means that side y in triangle CDF is equal to x 1, where x is the length of diagonals AC and AD. Since the triangles ACD and CDF are similar, we have the following relationship between the sides:
5.1) 
Simplifying this relation gives the following: 
Solving this quadratic equation via the quadratic formula gives the solution
This number is called the golden ratio or the golden mean
and is denoted by the symbol 
( the Greek letter phi -- pronounced fee).
So, x = 
and y = 
- 1. But from Equation 5.1) above, it can
be seen that
x 1 = 1/x and so
that 
- 1 = 1/
.
In triangle ABC we can see that 
.
So cos(36) = 
/2 = 
In triangle ABC we can see that h = sin(36).
And by using the theorem of Pythagoras, we also have
= 1 - 
From Equation 5.1) we can see that 
= 1 + 
= 1 + 
So 
Since h = sin(36),
5.2) sin(36)
= 
and
5.3) cos(36)
= 
We can check these results with a calculator:
sin(36) = 0.58778 and cos(36) = 0.80902
= 0.58778 
= 0.80902
It is also of interest to find the sine and cosine of the 108 degree angle at B in Figure 5.1. These values will be needed in the discussions of the dihedral angle of the dodecahedron and icosahedron which follow later in this document.
From the trigonometric identities
sin( 180 x ) = sin(x) = cos( 90 x )
we have
sin( 180 108 ) = sin(72) = cos( 18 )
sin( 72 ) = cos( 18 )
From the half-angle formula cos(x/2) = 
we have
cos(18) = 
But from Equation 5.3 above we know that
cos(36) = 
So cos(18) = 
So finally we have
cos( 18 ) = 
From the identities above we know that
sin(108) = sin( 72 ) = cos(18 ) and so
5.4) sin(
108 ) = 
To calculate the cosine of 108 degrees, we shall use the Law of Cosines.
The law may be stated as follows:
In any triangle, the square of any side is equal to the sum of the squares of the other two sides minus twice their product times the cosine of the included angle.
In Figure 6.1 refer to the triangle ABC. The Law of Cosines states the following
so
cos(108) = 
But x = F and F = 
So
cos(108) = 
But 
= 1 + 
so cos(108) = 1
- 
= 
And finally we have
5.5) cos(108)
= 
We can check these formulas with a calculator as follows:
sin( 108 ) = 0.9511 and cos( 108 ) = -0.30902
and 
and so the formulas check.
To calculate the radius of the circumscribing circle and the radius of the inscribed circle, refer to Figure 5.2.
In this figure, R is the radius of the circumscribing circle. That circle touches the pentagon at each of its vertices. Also in this figure, r is the radius of the inscribed circle. The inscribed circle touches the pentagon at the midpoints of each of its sides.
In this figure lines have been drawn from the centroid of the pentagon to each of the vertices and in one of the resulting triangles a line has been drawn from the centroid to the midpoint of one of the sides. From the figure, we can determine following relationships:
So, R = 
and r = 
R = 
This formula can be simplified by multiplying the numerator
and denominator by the quantity 
This gives R = 
R = 
And so,
5.6) R
= 
where R is the radius of the circumscribed circle.
Also from Figure 5.2, we can see that 
This follows from the theorem of Pythagoras.
Hence 
Substituting for R in this equation gives 
So we have finally that
5.7) r =
where r is the radius of the inscribed circle.
To find a and r in terms of R we start with Equation 5.6. This equation gives
= 0.80507a
So a = 1.17557 R. We can conveniently check our result against this number.
So, solving for a we get
So
To remove the 
x from the denominator of the radical we multiply the terms
under the radical by 
to get
To remove the 5 in the denominator of the radical, we
multiply the value under the radical by 
. This gives us
To remove the radical from the denominator, we multiply the
numerator and denominator by 
. This gives us the
following:
So finally we have
5.8) 
where a is the length of a side and R is the radius of the circumscribed circle.
To find r in terms of R we start with Equation 5.7.
Again we can conveniently check our result against this number.
We just showed in Equation 5.8 that 
Substituting the value for a in the equation for r we have
By multiplying the value under the second radical by 
we can remove the 
in the
denominator. Doing this we get
Simplifying this expression gives
We can simplify this further by realizing that
And so finally we have it!
5.9) 
where r is the radius of the inscribed circle and R is the radius of the circumscribed circle.
Figure 5.1
A regular pentagon with all vertices and angles marked. All sides are of length 1.
Reference [4]
Figure 5.2
Regular pentagon showing the inscribing radius, r, and the circumscribing radius, R.
Reference [4]