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 . Thus, you can write:

Solve for :

This type of problem can be solved through using the formula:

If the ratio of the hexagons is 4:5, then and . denotes area. The side length of the first hexagon is 16.

Substituting in the numbers, the formula looks like:

This quickly begins a problem where the second area can be solved for by rearranging all the given information.

The radical can then by simplified with or without a calculator.

Without a calculator, the 320 can be factored out and simplified:

This kind of problem can be solved for by using the formula:

where the values are the similarity ratio and the values are the side lengths.

In this case, the problem provides , but not . is denoted as in the question.

There are four variables in this formula, and three of them are provided in the problem. This means that we can solve for () by substituting in all known values and rearranging the formula so it's in terms of .

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.

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:

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:

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:

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!)

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:

Use the formula for perimeter to solve for the side length:

Use the formula for perimeter to solve for the side length:

Use the formula for perimeter to solve for the side length:

Use the formula for perimeter to solve for the length of a side of the regular hexagon:

Where is perimeter and is the length of a side.

In this case:

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

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 .