The area has no relevance to find the angle of a regular hexagon.

There are 6 sides in a regular hexagon.  Use the following formula to determine the interior angle.

Substitute  sides to determine the sum of all interior angles of the hexagon in degrees.

Since there are 6 sides, divide this number by 6 to determine the value of each interior angle.

Below is regular Hexagon  with center , a segment drawn from  to each vertex - that is, each of its radii drawn.

Each angle of a regular hexagon measures ; by symmetry, each radius bisects an angle of the hexagon, so 

.

The angles of a rectangle must measure , so it has been disproved that Quadrilateral  is a rectangle.

Suppose Hexagon  is regular. Each angle of a regular polygon of  sides has measure

A hexagon has 6 sides, so set ; each angle of the regular hexagon has measure  

Since one angle is given to be of measure , the hexagon cannot be regular.

The sum of the exterior angles of any polygon, one at each vertex, is . In a regular polygon, the exterior angles all have the same measure, so divide 360 by the number of angles, which, here, is 20, the same as the number of sides.

The sum of the interior angle measures of a hexagon is 

Add the angle measures in each group.

In each case, the angle measures add up to 720, so the answer is that all of these can be the six interior angle measures of a hexagon.

An equilateral hexagon can be divided into 6 equilateral triangles of side length 6.  

The area of a triangle is .  Since equilateral triangles have angles of 60, 60 and 60 the height is .  This gives each triangle an area of  for a total area of the hexagon at  or .

A regular hexagon can be divided into 12 30-60-90 right triangles with hypotenuse equal to the length of half of a diagonal, or 6 in this case (see image). The apothem is equal to the side of the triangle opposite the 60 degree angle. Therefore, based on the rules of a 30-60-90 right triangle, we conclude:

The area of a regular polygon can be calculated with the following formula:

The length of 1 side is half of the diagonal, or 6 in this case. The perimeter is the sum of lengths of the sides.

Therefore the area is equal to:

How do you find the area of a hexagon?

There are several ways to find the area of a hexagon.

1. In a regular hexagon, split the figure into triangles.
2. Find the area of one triangle.
3. Multiply this value by six.

Alternatively, the area can be found by calculating one-half of the side length times the apothem.

Regular hexagons:

Regular hexagons are interesting polygons. Hexagons are six sided figures and possess the following shape:

In a regular hexagon, all sides equal the same length and all interior angles have the same measure; therefore, we can write the following expression.

One of the easiest methods that can be used to find the area of a polygon is to split the figure into triangles. Let's start by splitting the hexagon into six triangles. 

In this figure, the center point, , is equidistant from all of the vertices. As a result, the six dotted lines within the hexagon are the same length. Likewise, all of the triangles within the hexagon are congruent by the side-side-side rule: each of the triangle's share two sides inside the hexagon as well as a base side that makes up the perimeter of the hexagon. In a similar fashion, each of the triangles have the same angles. There are  in a circle and the hexagon in our image has separated it into six equal parts; therefore, we can write the following:

We also know the following:

Now, let's look at each of the triangles in the hexagon. We know that each triangle has two two sides that are equal; therefore, each of the base angles of each triangle must be the same. We know that a triangle has  and we can solve for the two base angles of each triangle using this information.

Each angle in the triangle equals . We now know that all the triangles are congruent and equilateral: each triangle has three equal side lengths and three equal angles. Now, we can use this vital information to solve for the hexagon's area. If we find the area of one of the triangles, then we can multiply it by six in order to calculate the area of the entire figure. Let's start by analyzing . If we draw, an altitude through the triangle, then we find that we create two  triangles. 

Let's solve for the length of this triangle. Remember that in  triangles, triangles possess side lengths in the following ratio:

Now, we can analyze  using the a substitute variable for side length, .

We know the measure of both the base and height of  and we can solve for its area.

Now, we need to multiply this by six in order to find the area of the entire hexagon.

We have solved for the area of a regular hexagon with side length, . If we know the side length of a regular hexagon, then we can solve for the area.

If we are not given a regular hexagon, then we an solve for the area of the hexagon by using the side length(i.e. ) and apothem (i.e. ), which is the length of a line drawn from the center of the polygon to the right angle of any side. This is denoted by the variable  in the following figure:

Alternative method:

If we are given the variables  and , then we can solve for the area of the hexagon through the following formula:

In this equation,  is the area,  is the perimeter, and  is the apothem. We must calculate the perimeter using the side length and the equation , where  is the side length.

Solution:

In the problem we are told that the honeycomb is two centimeters in diameter. In order to solve the problem we need to divide the diameter by two. This is because the radius of this diameter equals the interior side length of the equilateral triangles in the honeycomb. Lets find the side length of the regular hexagon/honeycomb.

Substitute and solve.

We know the following information.

As a result, we can write the following:

Let's substitute this value into the area formula for a regular hexagon and solve.

Simplify.

Solve.

Round to the nearest tenth of a centimeter.