The sum of the interior angles of a polygon is given by where = number of sides of the polygon. An octagon has 8 sides, so the formula becomes

The sum of the interior angles of a hexagon must equal 720 degrees. Because the hexagon is regular, all of the interior angles will have the same measure. A hexagon has six sides and six interior angles. Therefore, each angle measures.

The sum of the degree measures of the angles of a (six-sided) hexagon, is

We can solve for in the equation

The sum of the degree measures of the angles of a (six-sided) hexagon, is

We can solve for in the equation

The sum of the degree measures of the angles of a (six-sided) hexagon, is

Let be the common measure of the three congruent angles in question. We can solve for in the equation

To find the sum of the interior angles of any regular polygon, use the formula , where represents the number of sides of the regular polygon.

The sum of the interior angles of a regular hexagon is 720 degrees. To find the measurement of one angle, divide by the number of interior angles (or sides):

The measurement of one angle in a regular hexagon is 120 degrees.

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 given problem we know that the side length of a regular hexagon is the following:

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

Simplify.

Round the answer to the nearest whole number.

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.

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 .

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 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 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 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:

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 :

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

The sum of the interior angles of a polygon is given by where = number of sides of the polygon. An octagon has 8 sides, so the formula becomes

The sum of the interior angles of a hexagon must equal 720 degrees. Because the hexagon is regular, all of the interior angles will have the same measure. A hexagon has six sides and six interior angles. Therefore, each angle measures.