A diagonal is a line segment joining two non-adjacent vertices of a polygon.  A regular hexagon has six sides and six vertices.  One vertex has three diagonals, so a hexagon would have three diagonals times six vertices, or 18 diagonals.  Divide this number by 2 to account for duplicate diagonals between two vertices. The formula for the number of vertices in a polygon is:

where .

A diagonal connects two non-consecutive vertices of a polygon.  A hexagon has six sides.  There are 3 diagonals from a single vertex, and there are 6 vertices on a hexagon, which suggests there would be 18 diagonals in a hexagon.  However, we must divide by two as half of the diagonals are common to the same vertices. Thus there are 9 unique diagonals in a hexagon. The formula for the number of diagonals of a polygon is:

where n = the number of sides in the polygon.

Thus a pentagon thas 5 diagonals.  An octagon has 20 diagonals.

The key is to examine  in thie figure below:

Each interior angle of a regular hexagon, including , measures , so it can be easily deduced by way of the Isosceles Triangle Theorem that . , so by angle addition, 

.

Also, by symmetry,

,

so ,

and  is a  triangle whose long leg  has length .

By the  Theorem, , which is the hypotenuse of , has length  times that of the long leg, so .

The key is to examine  in thie figure below:

Each interior angle of a regular hexagon, including , measures , so it can be easily deduced by way of the Isosceles Triangle Theorem that . , so by angle addition, 

.

Also, by symmetry,

,

so ,

and  is a  triangle whose hypotenuse  has length .

By the  Theorem, the long leg  of  has length  times that of hypotenuse , so .

The key is to examine  in thie figure below:

Each interior angle of a regular hexagon, including , measures , so it can be easily deduced by way of the Isosceles Triangle Theorem that . To find   we can subtract  from . Thus resulting in:

Also, by symmetry,

,

so .

Therefore,  is a  triangle whose short leg  has length  .

The hypotenuse  of this   triangle measures twice the length of short leg , so .

The key is to examine  in thie figure below:

Each interior angle of a regular hexagon, including , measures , so it can be easily deduced by way of the Isosceles Triangle Theorem that . To find  we subtract  from  . Thus resullting in 

Also, by symmetry,

,

so ,

and  is a  triangle whose short leg  has length .

The long leg  of this  triangle measures  times the length of short leg , so .

The perimeter of the figure can be expressed in terms of the variables by adding:

Simplify and set :

Along with , we now have a system of equations to solve for  by adding:

The perimeter of the figure can be expressed in terms of the variables by adding:

Simplify and set :

The perimeter of the figure can be expressed in terms of the variables by adding:

Simplify and set :

Since the ratio of  to  is equivalent to  - or

,

then 

and we can substitute as follows: