How to find the area of a hexagon - Geometry

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Question

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

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

Answer

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.

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:

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

$\theta=60^{\circ$

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.

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, .

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:

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 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.

$\textup{Diameter}=2\times \textup{ radius}$

Substitute and solve.

$2\textup{ cm}=2\times \textup{radius}$

$\textup{radius}=\frac{2\textup{ cm}}{2}$

$\textup{radius}=1\textup{ cm}$

We know the following information.

$\textup{radius}=\textup{side length}$

As a result, we can write the following:

$s=1\textup{ cm}$

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

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

$A=\frac{3\sqrt{3}}{2}\times (1\textup{ cm})^2$

Simplify.

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

Solve.

$A=2.59\textup{ cm}^2$

Round to the nearest tenth of a centimeter.

$A=2.6\textup{ cm}^2$

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Question

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

Answer

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.

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

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Question

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

Answer

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.

$\text{Area}=\frac{3\sqrt3}{2}(2^2)=\frac{3\sqrt3}{2}(4)=\frac{12\sqrt3}{2}=6\sqrt3$

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Question

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

Answer

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.

$\text{Area}=\frac{3\sqrt3}{2}(6^2)=\frac{3\sqrt3}{2}(36)=\frac{108\sqrt3}{2}=54\sqrt3$

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Question

A regular (also known as equilateral) hexagon has an apothem length of $7\sqrt{3}$ .

Find the area of the hexagon.

Answer

Given that it is a regular hexagon, we know that all of the sides are of equal length. Therefore, all 6 of the triangles (we get from drawing lines to opposite vertexes) are congruent triangles.

Lets use the fact that there are 360 degrees in a full rotation. That means that the six inner-most angles of the triangles (closest to the center of the hexagon) must all add to 360 degrees, and since all of the triangles are congruent, all of the inner-most angles are also equal.

$\frac{360}{6}=60^{\circ}$

If the innermost angle is 60 degrees, and the fact that it is a regular hexagon, we can therefore state that the other two angles are

$\frac{\left ( 180-60 \right )}{2}=60^{\circ}$

as well. Since all of the angles are 60 degrees, it is an equilateral triangle.

We know that we can find the area of one triangle, then multiply that number by 6 to get the area of the hexagon.

With that in mind, we look at the right triangle.

Look familiar?? The 30-60-90 and 45-45-90 right triangles appear often, so it's worth it to remember the shortcuts to the Pythagorean Theorem so you don't have to do the calculations every time.

For our numbers we have $\mathbf{a}\sqrt{3} = 7\sqrt{3}}$ , so our base is going to be 7 cm.

The base of the larger equilateral triangle is going to be twice as big, so

$7\cdot 2=14\ cm$ .

Plug these into

$A=base\cdot height\cdot \frac{1}{2}$

and then multiply by 6 (remember we want the area of all 6 triangles inside of the hexagon)

$14 \cdot 7\sqrt{3} \cdot \frac{1}{2} = 49\sqrt{3}}$

$49\sqrt{3} \cdot 6 = 294\sqrt{3}\approx 509.22$

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Question

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

Answer

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 .

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Question

What is the area of a regular hexagon with a long diagonal of length 12?

Answer

A regular hexagon can be divided into 12 30-60-90 right triangles with hypotenuse equal to the length of half of a diagonal, or 6 in this case (see image). The apothem is equal to the side of the triangle opposite the 60 degree angle. Therefore, based on the rules of a 30-60-90 right triangle, we conclude:

$apothem=x\sqrt3=3\sqrt3$

The area of a regular polygon can be calculated with the following formula:

$A=\frac{1}{2}(apothem)(perimeter)$

The length of 1 side is half of the diagonal, or 6 in this case. The perimeter is the sum of lengths of the sides.

Therefore the area is equal to:

$A=\frac{1}{2}(3\sqrt3)(36)=54\sqrt3$

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Question

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

Answer

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.

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

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Question

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

Answer

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.

$\text{Area}=\frac{3\sqrt3}{2}(10^2)=\frac{3\sqrt3}{2}(100)=\frac{300\sqrt3}{2}=150\sqrt3$

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Question

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

Answer

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.

$\text{Area}=\frac{3\sqrt3}{2}(7^2)=\frac{3\sqrt3}{2}(49)=\frac{147\sqrt3}{2}$

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Question

Find the area of a regular hexagon with a side length of $\frac{1}{2}$ .

Answer

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.

$\text{Area}=\frac{3\sqrt3}{2}\left(\frac{1}{2}\right)^2=\frac{3\sqrt3}{2}\left(\frac{1}{4}\right)=\frac{3\sqrt3}{8}$

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Question

Find the area of a regular hexagon with side lengths of $\frac{1}{3}$ .

Answer

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.

$\text{Area}=\frac{3\sqrt3}{2}\left(\frac{1}{3}\right)^2=\frac{3\sqrt3}{2}\left(\frac{1}{9}\right)=\frac{3\sqrt3}{18}=\frac{\sqrt3}{6}$

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Question

In terms of , find the area of a regular hexagon with side lengths of .

Answer

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.

$\text{Area}=\frac{3\sqrt3}{2}(2x)^2=\frac{3\sqrt3}{2}(4x^2)=\frac{12x^2\sqrt3}{2}=6x^2\sqrt3$

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Question

In terms of , find the area of a regular hexagon with side lengths .

Answer

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.

$\text{Area}=\frac{3\sqrt3}{2}(12x)^2=\frac{3\sqrt3}{2}(144x^2)=\frac{432x^2\sqrt3}{2}=216x^2\sqrt3$

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Question

In terms of , find the area of a regular hexagon with side lengths .

Answer

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.

$\text{Area}=\frac{3\sqrt3}{2}(9x)^2=\frac{3\sqrt3}{2}(81x^2)=\frac{243x^2\sqrt3}{2}$

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Question

In terms of , find the area of the regular hexagon with side lengths .

Answer

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.

$\text{Area}=\frac{3\sqrt3}{2}(8x)^2=\frac{3\sqrt3}{2}(64x^2)=\frac{192x^2\sqrt3}{2}=96x^2\sqrt3$

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Question

A hexagon can be divided into six congruent equilateral triangles. The triangles have a sides that are 10 inches long. What is the area of the hexagon?

Answer

To find the area of the hexagon we need to find the area of one of the equilateral triangles then multiply by six. To do this we have to find the height of one of the triangles. We find the height by using the Pythagorean theorem:

For one of our six equilateral triangles this looks like:

where 5 is half of one of our sides and b is the height of of the triangle (remember in an equilateral triangle the height will bisect the opposite side). We can now solve the equation for the missing hieght, b.

$25 + b^2 = 100 \rightarrow b^2 = 75 \text ; \rightarrow b = \sqrt 75 \rightarrow ewline b= \sqrt25\sqrt 3 \rightarrow b= 5\sqrt3$

So the height of one of our triangles is 5 times the square root of three. Now we have all the information we need. Using the triangle area formula, base times height divided by two we get:

$\frac{1}{2}(b x h) = \frac{1}{2}(10\times 5\sqrt3 )= \frac{1}{2}(50\sqrt3 ) = 25\sqrt3 ; in^2$

This is the area for ONE of our triangles, but we have six! So we will just multiply this area by six:

$25\sqrt3 ; in^2 \times 6 = 150\sqrt3 ; in^2$

This is our final answer.

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Question

If a regular hexagon has a side length of , what would be the area? Assume all sides are of equal length and round to the nearest tenth.

Answer

The following equation can be used to solve for area of a regular polygon, given the side length and number of sides :

$\Area=\frac{s^2N}{4tan(\frac{180^{\circ}}{N})}\ \ \Area=\frac{(8.7;cm)^2(6)}{4tan(\frac{180^{\circ}}{6})}$

$Area=\frac{454.14;cm^2}{2.3094}\approx196.7;cm^2$

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Question

A regular hexagon has a side length of . Assuming all sides are of equal length, what is the area of the hexagon rounded to the nearest tenth?

Answer

We can solve for area using the given formula that works for all regular polygons, with being the side length and representing the number of sides:

$\Area=\frac{s^2N}{4tan(\frac{180^{\circ}}{N})}\ \ \Area=\frac{(7.6;cm)^2(6)}{4tan(\frac{180^{\circ}}{6})}$

$Area=\frac{346.56;cm^2}{2.3094}\approx150.1;cm^2$

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Question

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.

Answer

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.

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

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