High School Math : How to find the area of a hexagon

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #1 : How To Find The Area Of A Hexagon

Calculate the approximate area a regular hexagon with the following side length:

Possible Answers:

Cannot be determined 

Correct answer:

Explanation:

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. 

Screen shot 2016 07 06 at 2.09.44 pm

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:

Screen shot 2016 07 06 at 2.27.41 pm

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. 

Screen shot 2016 07 06 at 2.27.10 pm

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, .

Screen shot 2016 07 06 at 3.01.03 pm

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:

Screen shot 2016 07 06 at 3.17.05 pm

 

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.

Example Question #1 : How To Find The Area Of A Hexagon

A single hexagonal cell of a honeycomb is two centimeters in diameter.

Screen shot 2016 07 06 at 4.46.18 pm 

What’s the area of the cell to the nearest tenth of a centimeter?

 

Possible Answers:

Cannot be determined 

Correct answer:

Explanation:

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. 

Screen shot 2016 07 06 at 2.09.44 pm

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:

Screen shot 2016 07 06 at 2.27.41 pm

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. 

Screen shot 2016 07 06 at 2.27.10 pm

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, .

Screen shot 2016 07 06 at 3.01.03 pm

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:

Screen shot 2016 07 06 at 3.17.05 pm

 

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.

Example Question #2 : How To Find The Area Of A Hexagon

What is the area of a regular hexagon with an apothem of  and a side length of ?

Possible Answers:

Correct answer:

Explanation:

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. 

Screen shot 2016 07 06 at 2.09.44 pm

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:

Screen shot 2016 07 06 at 2.27.41 pm

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. 

Screen shot 2016 07 06 at 2.27.10 pm

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, .

Screen shot 2016 07 06 at 3.01.03 pm

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:

Screen shot 2016 07 06 at 3.17.05 pm

 

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 a hexagon the number of sides is  and in this example the side length is .

The perimeter is .

Then we plug in the numbers for the apothem and perimeter into the original equation.

The area is .

Example Question #1 : How To Find The Area Of A Hexagon

Hexagon

This provided figure is a regular hexagon with a side length with the following measurement:

Calculate the area of the regular hexagon.

Possible Answers:

Correct answer:

Explanation:

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. 

Screen shot 2016 07 06 at 2.09.44 pm

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:

Screen shot 2016 07 06 at 2.27.41 pm

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. 

Screen shot 2016 07 06 at 2.27.10 pm

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, .

Screen shot 2016 07 06 at 3.01.03 pm

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:

Screen shot 2016 07 06 at 3.17.05 pm

 

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.

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