Because we're told the pentagon is a regular polygon, this means that all of its sides are the same length. That is, the side lengths are congruent.

In order to solve for the perimeter, which is the sum of all sides, we can use a multiplication shortcut of:

, where  is perimeter,  is length of the sides, and  is the number of sides of the polygon. 

Using the information given in the problem, the values can be substituted in.

Note that this math is the same as adding  five times:

There are two key things for a question like this. The first is to know that a polygon has a total degree measure of:

, where  is the number of sides.

Therefore, a hexagon like this one has:

. 

Next, you should remember that all of the exterior angles listed are supplementary to their correlative interior angles. This lets you draw the following figure: 

Now, you just have to manage your algebra well. You must sum up all of the interior angles and set them equal to . Thus, you can write:

Solve for :

There are two key things for a question like this. The first is to know that a polygon has a total degree measure of:

, where  is the number of sides.

Therefore, a hexagon like this one has:

.

Next, you should remember that all of the exterior angles listed are supplementary to their correlative interior angles. This lets you draw the following figure: 

Now, you just have to manage your algebra well. You must sum up all of the interior angles and set them equal to . Thus, you can write:

Solve for :

Now, you have to find the largest unknown angle, which is :

The sum of all angles is determined by the following formula for a polygon:

In a pentagon, there are 5 sides, or .  Substitute and find the total possible angle in a pentagon.

There are 5 interior angles in a pentagon.  Divide the total possible angle by 5 to determine the value of one interior angle.

Each interior angle of a pentagon is 108 degrees.

The sum of three angles in a pentagon is:

You can redraw the figure given to notice the little equilateral triangle that is formed within the hexagon. Since a hexagon can have the  degrees of its internal rotation divided up evenly, the central angle is  degrees. The two angles formed with the sides also are  degrees. Thus, you could draw:

Now, the  is located on the side that is the same as  on your standard  triangle. The base of the little triangle formed here is  on the standard triangle. Let's call our unknown value .

We know, then, that:

Or, 

Now, this is only half of the size of the hexagon's side. Therefore, the full side length is .

Since this is a regular hexagon, all of the sides are of equal length.  This means that your total perimeter is  or .

The area of a regular hexagon is defined by the equation:

, where  is the length of a side.

This is derived from the fact that the regular hexagon can be split up into  little equilateral triangles, each having an area of 

To visualize this, consider the drawing:

Each triangle formed like this will be equilateral.  It is easiest to remember this relationship and memorize the general area equation for equilateral triangles. (It is useful in many venues!)

So, for your data, you know:

Solving for , you get:

This means that 

Therefore, the perimeter of the figure is equal to  or .

There are 6 sides in a hexagon.  

Therefore, given a side length of 16, the perimeter is:

Write the formula for the perimeter of a hexagon.  

Substitute the given length.

To begin, calculate the side length of the hexagon. Since it is regular, its sides are of equal length. This means that a given side is  or  in length. Now, consider your figure like this: 

The little triangle at the top forms an equilateral triangle. This means that all of its sides are . You could form six of these triangles in your figure if you desired. This means that the long diagonal is really just  or .

You could redraw your figure as follows.  Notice that this kind of figure makes an equilateral triangle within the hexagon.  This allows you to create a useful  triangle.

The  in the figure corresponds to  in a reference  triangle. The hypotenuse is  in the reference triangle. 

Therefore, we can say:

Solve for :

Rationalize the denominator:

Now, the diagonal of a regular hexagon is actually just double the length of this hypotenuse. (You could draw another equilateral triangle on the bottom and duplicate this same calculation set—if you wanted to spend extra time without need!) Thus, the length of the diagonal is: