How to find the area of a hexagon

Help Questions

Math › How to find the area of a hexagon

Questions 1 - 10

An equilateral hexagon has sides of length 6, what is it's area?

$54\sqrt{3}$

$36\sqrt{3}$

$96\sqrt{3}$

Explanation

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

The area of a triangle is $0.5\cdot b\cdot h$ . Since equilateral triangles have angles of 60, 60 and 60 the height is $\frac{6\sqrt{3}}{2}$ . This gives each triangle an area of $9\sqrt{3}$ for a total area of the hexagon at $54\sqrt{3}$ or .

Find the area of a regular hexagon with a side length of .

$24\sqrt3$

$48\sqrt3$

$8\sqrt3$

$36\sqrt3$

Explanation

Use the following formula to find the area of a regular hexagon:

$\text{Area}=\frac{3\sqrt3}{2}side^2$

Now, substitute in the value for the side length.

$\text{Area}=\frac{3\sqrt3}{2}(4^2)=\frac{3\sqrt3}{2}(16)=\frac{48\sqrt3}{2}=24\sqrt3$

A circle is placed inside a regular hexagon as shown in the figure.

If the radius of the circle is $\frac{2}{3}$ , then find the area of the shaded region.

Explanation

In order to find the area of the shaded region, we must first find the areas of the hexagon and the circle.

Recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side})^2$

Substitute in the length of the given side to find the area of the hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(4)^2$

$\text{Area of Regular Hexagon}=24\sqrt3$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

Substitute in the given radius to find the area of the circle.

$\text{Area of Circle}=\pi\times \left(\frac{2}{3}\right)^2$

$\text{Area of Circle}=\frac{4}{9}\pi$

Now, find the area of the shaded region by subtracting the area of the circle from the area of the hexagon.

$\text{Area of Shaded Region}=\text{Area of Hexagon}-\text{Area of Circle}$

$\text{Area of Shaded Region}=24\sqrt3-\frac{4}{9}\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=40.17$

A rectangle is attached to a regular hexagon as shown by the figure.

If the length of the diagonal of the hexagon is , find the area of the entire figure.

Explanation

When all of the diagonals of a regular hexagon are drawn in, you should notice that the hexagon is divided into six congruent equilateral triangles. The length of the diagonal is twice the length of a side of one of the equilateral triangles.

Thus, the diagonal of a hexagon is also twice the length of a side of the regular hexagon.

$\text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$

Substitute in the given diagonal to find the side length of the hexagon.

$\text{Side Length of Hexagon}=\frac{14}{2}=7$

Now, recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3(\text{side length})^2}{2}$

Substitute in the value of the side length to find the area of the hexagon.

$\text{Area of Hexagon}=\frac{3\sqrt3(7)^2}{2}=\frac{147\sqrt3}{2}$

Next, notice that the hexagon shares a side with the length of the rectangle.

Recall how to find the area of a rectangle.

$\text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the length and the width of the rectangle to find the area.

$\text{Area of Rectangle}=7\times12=84$ .

In order to find the area of the entire figure, add the areas of the hexagon and rectangle together.

$\text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$

$\text{Area of Entire Figure}=\frac{147\sqrt3}{2}+84$

Solve and round to two decimal places.

$\text{Area of Entire Figure}=211.31$

In the regular hexagon above, if the length of diagonal is $3\sqrt3$ , what is the area of the hexagon?

Explanation

When all the diagonals of a hexagon are drawn in, you should notice that the long diagonals, marked by solid lines, form congruent equilateral triangles. You should also notice that the other diagonals drawn in with dashed lines are also the heights of two of the equilateral triangles.

Start by using the diagonal to find the length of a side of the hexagon.

Cut the diagonal in half so that we are just left with the height of one equilateral triangle. Notice that the height cleaves the equilateral triangle into two congruent triangles whose sides are in the ratios of $1:\sqrt3:2$ .

Set up a proportion to solve for the length of a side of the triangle.

$\frac{\text{side length}}{2}=\frac{\frac{\text{diagonal}}{2}}{\sqrt3}$

$\text{side length}=\frac{\text{diagonal}}{\sqrt3}$

Plug in the given diagonal to solve for the side length.

$\text{side length}=\frac{3\sqrt3}{\sqrt3}=3$

Now, recall how to find the area of a regular hexagon:

$\text{Area of Hexagon}=\frac{3\sqrt3}{2}(\text{side})^2$

Plug in the side length that you just found in order to find the area.

$\text{Area of Hexagon}=\frac{3\sqrt3}{2}(3)^2=23.38$

Make sure to round to places after the decimal.

A circle is placed inside a regular hexagon as shown in the figure.

If the radius of the circle is , then find the area of the shaded region.

Explanation

In order to find the area of the shaded region, we must first find the areas of the hexagon and the circle.

Recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side})^2$

Substitute in the length of the given side to find the area of the hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(6)^2$

$\text{Area of Regular Hexagon}=54\sqrt3$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

Substitute in the given radius to find the area of the circle.

$\text{Area of Circle}=\pi\times (2)^2$

$\text{Area of Circle}=4\pi$

Now, find the area of the shaded region by subtracting the area of the circle from the area of the hexagon.

$\text{Area of Shaded Region}=\text{Area of Hexagon}-\text{Area of Circle}$

$\text{Area of Shaded Region}=54\sqrt3-4\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=80.96$

Hexagon

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

$s=6\textup{ cm}$

Calculate the area of the regular hexagon.

$93.53\textup{ cm}^{2}$

$61.94\textup{ cm}^{2}$

$216\textup{ cm}^{2}$

$36\textup{ cm}^{2}$

$72\textup{ cm}^{2}$

Explanation

How do you find the area of a hexagon?

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

In a regular hexagon, split the figure into triangles.
Find the area of one triangle.
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.

$\overline{AB}=\overline{BC}=\overline{CD}=\overline{DE}=\overline{EF}=\overline{FA}$

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 $360^{\circ}$ 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

$\theta=\frac{360^{\circ}}{6}$

$\theta=60^{\circ$

We also know the following:

$\angle AXB=\angle BXC=\angle CXD=\angle DXE=\angle EXF=\angle FXA=60^{\circ}$

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 $180^{\circ}$ and we can solve for the two base angles of each triangle using this information.

$2x+60^{\circ}=180^{\circ}$

$2x+=120^{\circ}$

$x=60^{\circ}$

Each angle in the triangle equals $60^{\circ}$ . 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 $\triangle BXC$ . If we draw, an altitude through the triangle, then we find that we create two $30^{\circ}-60^{\circ}-90^{\circ}$ triangles.

Screen shot 2016 07 06 at 2.27.10 pm

Let's solve for the length of this triangle. Remember that in $30^{\circ}-60^{\circ}-90^{\circ}$ triangles, triangles possess side lengths in the following ratio:

$1:2:\sqrt{3}$

Now, we can analyze $\triangle BXC$ 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 $\triangle BXC$ and we can solve for its area.

$\textup{Area of }\triangle BXC=\frac{1}{2}\times \textup{base}\times \textup{height}$

$\textup{Area of }\triangle BXC=\frac{1}{2}\times s\times \frac{\sqrt{3}s}{2}$

$\textup{Area of }\triangle BXC=\frac{\sqrt{3}s^2}{4}$

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

$\textup{Area of Hexagon}[ABCDEF]=6\times\frac{\sqrt{3}s^2}{4}$

$\textup{Area of Hexagon}[ABCDEF]=\frac{6 \sqrt{3}s^2}{4}$

$\textup{Area of Hexagon}[ABCDEF]=\frac{3\sqrt{3}}{2}\times s^2$

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:

$A=\frac{1}{2}(P)(a)$

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 $P=\textup{number of sides}\times s$ , where is the side length.

Solution:

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

$s=6\textup{ cm}$

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

$A=\frac{3\sqrt{3}}{2}\times s^2$

$A=\frac{3\sqrt{3}}{2}\times (6\textup{ cm})^2$

Simplify.

$A=\frac{108\sqrt{3}}{2} \textup{ cm}^{2}$

$A=54\sqrt{3} \textup{ cm}^{2}$

$A=93.53 \textup{ cm}^{2}$

A rectangle is attached to a regular hexagon as shown by the figure.

If the diagonal of the hexagon is , find the area of the entire figure.

Explanation

Thus, the diagonal of a hexagon is also twice the length of a side of the regular hexagon.

$\text{Side Length of Hexagon}=\frac{\text{Diagonal of Hexagon}}{2}$

Substitute in the given diagonal to find the side length of the hexagon.

$\text{Side Length of Hexagon}=\frac{10}{2}=5$

Now, recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3(\text{side length})^2}{2}$

Substitute in the value of the side length to find the area of the hexagon.

$\text{Area of Hexagon}=\frac{3\sqrt3(5)^2}{2}=\frac{75\sqrt3}{2}$

Next, notice that the hexagon shares a side with the length of the rectangle.

Recall how to find the area of a rectangle.

$\text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the length and the width of the rectangle to find the area.

$\text{Area of Rectangle}=5\times 8=40$ .

In order to find the area of the entire figure, add the areas of the hexagon and rectangle together.

$\text{Area of Entire Figure}=\text{Area of Hexagon}+\text{Area of Rectangle}$

$\text{Area of Entire Figure}=\frac{75\sqrt3}{2}+40$

Solve and round to two decimal places.

$\text{Area of Entire Figure}=104.95$

Find the area of a regular hexagon with a side length of .

$\frac{75\sqrt3}{2}$

$\frac{5\sqrt3}{2}$

$5\sqrt3$

$25\sqrt3$

Explanation

Use the following formula to find the area of a regular hexagon:

$\text{Area}=\frac{3\sqrt3}{2}side^2$

Now, substitute in the value for the side length.

$\text{Area}=\frac{3\sqrt3}{2}(5^2)=\frac{3\sqrt3}{2}(25)=\frac{75\sqrt3}{2}$

A circle is placed in a regular hexagon as shown in the figure below.

If the radius of the circle is , then find the area of the shaded region.

Explanation

In order to find the area of the shaded region, we must first find the areas of the hexagon and the circle.

Recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side})^2$

Substitute in the length of the given side to find the area of the hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(9)^2$

$\text{Area of Regular Hexagon}=\frac{243\sqrt3}{2}$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

Substitute in the given radius to find the area of the circle.

$\text{Area of Circle}=\pi\times (4)^2$

$\text{Area of Circle}=16\pi$

Now, find the area of the shaded region by subtracting the area of the circle from the area of the hexagon.

$\text{Area of Shaded Region}=\text{Area of Hexagon}-\text{Area of Circle}$

$\text{Area of Shaded Region}=\frac{243\sqrt3}{2}-16\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=160.18$

Page 1 of 6

Return to subject

AI TutorYour personal study buddy

Questions of the DayDaily practice to build your skills

FlashcardsStudy and memorize key concepts

WorksheetsBuild custom practice quizzes

AI SolverStep-by-step problem solutions

Learn by ConceptMaster concepts step by step

Diagnostic TestsAssess your knowledge level

Practice TestsTest your skills with exam questions

GamesLearn through free games