ACT Math : How to find the area of a hexagon

Study concepts, example questions & explanations for ACT Math

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Example Questions

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

A hexagon is made up of 6 congruent equilateral triangles. Each equilateral triangle has a length of 8 units. What is the area in square units of the hexagon?

Possible Answers:

Correct answer:

Explanation:

First, let's draw out the hexagon.

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.

Equitr

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.

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

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

Possible Answers:

Correct answer:

Explanation:

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

Hex6

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

Hexagontriangle

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:

Example Question #52 : Plane Geometry

Hexcenter51

The figure above is a regular hexagon.  is the center of the figure. The line drawn is perpendicular to the side.

What is the area of the figure above?

Possible Answers:

Correct answer:

Explanation:

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:

Hexcenter52

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:

 

Example Question #3 : How To Find The Area Of A Hexagon

What is the area of a regular hexagon with a perimeter of ?

Possible Answers:

Correct answer:

Explanation:

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:

Hex12

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

Hexagontriangle

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:

Example Question #53 : Act Math

What is the area of a regular hexagon with a side length of  miles? Simplify all fractions and square roots in your answer.

Possible Answers:

 

Correct answer:

 

Explanation:

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.

Example Question #54 : Act Math

What is the area of a hexagon with a side of length two? Simplify all fractions and square roots.

Possible Answers:

Correct answer:

Explanation:

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!)

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.

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