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:




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 +




Since h = sin(36),


5.2)      sin(36) =




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





cos(108) =


But x = F         and  F  =





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 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.




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








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




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!





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]