Hexagons - Intermediate Geometry

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Question

What is the measure of one exterior angle of a regular twenty-sided polygon?

Answer

The sum of the exterior angles of any polygon, one at each vertex, is $360^{\circ }$ . In a regular polygon, the exterior angles all have the same measure, so divide 360 by the number of angles, which, here, is 20, the same as the number of sides.

$360\div 20 = 18$

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Question

Which of the following cannot be the six interior angle measures of a hexagon?

Answer

The sum of the interior angle measures of a hexagon is $180 (6-2)=720^{\circ }$

Add the angle measures in each group.

In each case, the angle measures add up to 720, so the answer is that all of these can be the six interior angle measures of a hexagon.

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Question

There is a regular hexagon with a side length of . What is the measure of an internal angle?

Answer

Given that the hexagon is a regular hexagon, this means that all the side length are congruent and all internal angles are congruent. The question requires us to solve for the measure of an internal angle. Given the aforementioned definition of a regular polygon, this means that there must only be one correct answer.

In order to solve for the answer, the question provides additional information that isn't necessarily required. The measure of an internal angle can be solved for using the equation:

$\frac{180^{\circ}(n-2)}{n}$ where is the number of sides of the polygon.

In this case, .

For this problem, the information about the side length may be negated.

$\frac{180^{\circ}(6-2)}{6}$

$\frac{180^{\circ}(4)}{6}$

$\frac{180^{\circ}(2)}{3}$

$120^{\circ}$

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Question

What is the interior angle of a regular hexagon if the area is 15?

Answer

The area has no relevance to find the angle of a regular hexagon.

There are 6 sides in a regular hexagon. Use the following formula to determine the interior angle.

$\theta=(n-2)\cdot 180$

Substitute sides to determine the sum of all interior angles of the hexagon in degrees.

$\sum\theta=(6-2)\cdot 180=720$

Since there are 6 sides, divide this number by 6 to determine the value of each interior angle.

$\frac{720}{6}=120$

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Question

Given: Regular Hexagon with center . Construct segments $\overline{CA}$ and $\overline{CO}$ to form Quadrilateral .

True or false: Quadrilateral is a rectangle.

Answer

Below is regular Hexagon with center , a segment drawn from to each vertex - that is, each of its radii drawn.

Each angle of a regular hexagon measures $120^{\circ }$ ; by symmetry, each radius bisects an angle of the hexagon, so

$m \angle CAG = 60 ^{\circ }$ .

The angles of a rectangle must measure $90^{\circ }$ , so it has been disproved that Quadrilateral is a rectangle.

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Question

True or false: Each of the six angles of a regular hexagon measures $120^{\circ }$ .

Answer

A regular polygon with sides has congruent angles, each of which measures

$\frac{(N-2)180^{\circ }}{N}$

Setting , the common angle measure can be calculated to be

$\frac{(6-2)180^{\circ }}{6} = \frac{4 \cdot 180^{\circ }}{6} = \frac{720^{\circ }}{6} = 120^{\circ }$

The statement is therefore true.

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Question

True or false: Each of the exterior angles of a regular hexagon measures $80^{\circ }$ .

Answer

If one exterior angle is taken at each vertex of any polygon, and their measures are added, the sum is $360^{\circ }$ . Each exterior angle of a regular hexagon has the same measure, so if we let be that common measure, then

$6t = 360 ^{\circ }$

Solve for :

$6t \div 6 = 360 ^{\circ } \div 6$

$t = 60^{\circ }$

The statement is false.

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Question

Given: Hexagon .

$m \angle H = 108 ^{\circ }$

True, false, or undetermined: Hexagon is regular.

Answer

Suppose Hexagon is regular. Each angle of a regular polygon of sides has measure

$\frac{(N-2)180^{\circ }}{N}$

A hexagon has 6 sides, so set ; each angle of the regular hexagon has measure

$\frac{(6-2)180^{\circ }}{6} = \frac{ 4 \cdot 180^{\circ }}{6} = 120^{\circ }$

Since one angle is given to be of measure $108 ^{\circ }$ , the hexagon cannot be regular.

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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

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.

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

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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

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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