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.

A regular pentagon with all vertices and angles marked. All sides are of length 1.

Reference [4]

Regular pentagon showing the inscribing radius, r, and the circumscribing radius, R.

Reference [4]

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