To find the side of a hexagon given the area, set the area formula equal to the given area and solve for the side.

In a regular hexagon, all of the sides are the same length, and all of the angles are equivalent. The problem tells us that all of the angles inside the hexagon sum to . To find the value of one angle, we must divide  by , since there are  angles inside of a hexagon. 

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 . Since there are  angles, you know that the numeric portion will be  or . Thus, you can write:

Simplify and solve for :

This is  or .

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 :

First, let's draw out the hexagon.

Because the hexagon is made up of 6 equilateral triangles, to find the area of the hexagon, we will first find the area of each equilateral triangle then multiply it by 6.

Using the Pythagorean Theorem, we find that the height of each equilateral triangle is .

The area of the triangle is then

Multiply this value by 6 to find the area of the hexagon.

This question is asking about the area of a regular hexagon that looks like this:

Now, you could proceed by noticing that the hexagon can be divided into little equilateral triangles:

By use of the properties of isosceles and  triangles, you could compute that the area of one of these little triangles is:

, where  is the side length. Since there are  of these triangles, you can multiply this by  to get the area of the regular hexagon:

It is likely easiest merely to memorize the aforementioned equation for the area of an equilateral triangle. From this, you can derive the hexagon area equation mentioned above. Using this equation and our data, we know:

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:

Another way to write  is:

Now, there are several ways you could proceed from here. Notice that there are  of those little triangles in the hexagon. Since you know that the are of a triangle is:

and for your data...

The area of the whole figure is:

A hexagon has  sides. A regular polygon is one that has sides that are of equal length. Therefore, if the side length of our polygon is taken to be , we know:

, or 

This question is asking about the area of a regular hexagon that looks like this:

Now, you could proceed by noticing that the hexagon can be divided into little equilateral triangles:

By use of the properties of isosceles and  triangles, you could compute that the area of one of these little triangles is:

, where  is the side length. Since there are  of these triangles, you can multiply this by  to get the area of the regular hexagon:

It is likely easiest merely to memorize the aforementioned equation for the area of an equilateral triangle. From this, you can derive the hexagon area equation mentioned above. Using this equation and our data, we know:

For a hexagon with side length , the formula for the area is 
.
We have a side length of 4 miles, so we plug that into the equation and simplify the fraction.

To find the area of a hexagon with a given side length, , use the formula:

Plugging in 2 for  and reducing we get:

. (remember order of operations, square first!)