Hexagons - Math

Card 1 of 64

0

Didn't Know

0

Didn't Know

Knew It

0

1 of 2019 left

Question

How many lines of symmetry can be found in a regular hexagon?

Tap to reveal answer

Answer

The number of lines of symmetry through a regular polygon is equal to the number of sides.

A hexagon has lines of symmetry through each vertex, giving three lines of symmetry that each connect two opposite vertices. The other three lines pass through the midpoints of opposite sides of the hexagon.

← Didn't Know|Knew It →

Question

Tap to reveal answer

Answer

← Didn't Know|Knew It →

Question

How many diagonals are there in a regular hexagon?

Tap to reveal answer

Answer

A diagonal connects two non-consecutive vertices of a polygon. A hexagon has six sides. There are 3 diagonals from a single vertex, and there are 6 vertices on a hexagon, which suggests there would be 18 diagonals in a hexagon. However, we must divide by two as half of the diagonals are common to the same vertices. Thus there are 9 unique diagonals in a hexagon. The formula for the number of diagonals of a polygon is:

$\frac{n\cdot (n-3)}{2}$

where n = the number of sides in the polygon.

Thus a pentagon thas 5 diagonals. An octagon has 20 diagonals.

← Didn't Know|Knew It →

Question

If a triangle has 180 degrees, what is the sum of the interior angles of a regular octagon?

Tap to reveal answer

Answer

The sum of the interior angles of a polygon is given by where = number of sides of the polygon. An octagon has 8 sides, so the formula becomes

← Didn't Know|Knew It →

Question

What is the measure of an interior angle of a regular hexagon?

Tap to reveal answer

Answer

The measure of an internal angle of an regular polygon can be determined using the following equation, where equals the number of sides:

$\angle =\frac{180(n-2)}{n}$

$\angle=\frac{180(6-2)}{6}=\frac{180(4)}{6}=120$

← Didn't Know|Knew It →

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?

Tap to reveal answer

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.

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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 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.

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

$A=\frac{3\sqrt{3}}{2}\times 6^2$

Simplify.

$A=\frac{108\sqrt{3}}{2}$

$A=54\sqrt{3}$

Round the answer to the nearest whole number.

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

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

$P=6\times7=42$

The perimeter is .

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

$A=\frac{1}{2}(P)(a)=\frac{1}{2}(42)(4)=(21)(4)=84$

The area is .

← Didn't Know|Knew It →

Question

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.

Tap to reveal answer

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 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.

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

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

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

← Didn't Know|Knew It →

Question

How many diagonals are there in a regular hexagon?

Tap to reveal answer

Answer

A diagonal is a line segment joining two non-adjacent vertices of a polygon. A regular hexagon has six sides and six vertices. One vertex has three diagonals, so a hexagon would have three diagonals times six vertices, or 18 diagonals. Divide this number by 2 to account for duplicate diagonals between two vertices. The formula for the number of vertices in a polygon is:

$\frac{n\cdot (n-3)}{2}$

where $n=#\ of\ sides$ .

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

Regular hexagons are comprised of six equilateral triangles; in our question, these triangles each have side length 4 (see diagram).

The length of a diagonal is equal to two times the length of the side. In this instance, the answer is 8.

← Didn't Know|Knew It →

Question

How many degrees is the interior angle of any regular hexagon?

Tap to reveal answer

Answer

$interior\ angle=\frac{(n-2)180^o}{n}$

To find the angle of any regular polygon, you find the number of sides, . In this example .

$interior\ angle=\frac{(n-2)180^o}{n}=\frac{(6-2)*180^o}{6}$

You then subtract from the number of sides, yielding .

Take and multiply it by to yield the total number of degrees in the regular hexagon.

Then, to find one individual angle, we divide by the total number of angles, .

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

The answer is .

← Didn't Know|Knew It →

Question

What is the perimeter of a hexagon with a side length of ?

Tap to reveal answer

Answer

To find the perimeter of a regular polygon, we take the length of each side, , and multiply it by the number of sides, .

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

The perimeter is .

← Didn't Know|Knew It →

Question

What is the perimeter of a regular hexagon with a side length of 12?

Tap to reveal answer

Answer

To find the perimeter of a regular hexagon you must first know the number of sides in a hexagon which is 6.

When you know the number of sides of a regular polygon to find the perimeter you must multiply the side length by the number of sides.

In this case it is

The answer for the perimeter is .

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

To find the perimeter of a regular hexagon you must first know the number of sides in a hexagon which is .

When you know the number of sides of a regular polygon to find the perimeter you must multiply the side length by the number of sides.

In this case it is

The answer for the perimeter is .

← Didn't Know|Knew It →

Question

This figure is a regular hexagon with one side measuring 6 cm.

What is the perimeter of the regular hexagon in centimeters?

Tap to reveal answer

Answer

The perimeter of a regular hexagon is just the sum of all 6 sides. Because it's a regular hexagon, the perimemter is just six times one side (6 cm) or 36 cm.

← Didn't Know|Knew It →

Question

Tap to reveal answer

Answer

← Didn't Know|Knew It →

Question

If a triangle has 180 degrees, what is the sum of the interior angles of a regular octagon?

Tap to reveal answer

Answer

The sum of the interior angles of a polygon is given by where = number of sides of the polygon. An octagon has 8 sides, so the formula becomes

← Didn't Know|Knew It →

Question

What is the measure of an interior angle of a regular hexagon?

Tap to reveal answer

Answer

The measure of an internal angle of an regular polygon can be determined using the following equation, where equals the number of sides:

$\angle =\frac{180(n-2)}{n}$

$\angle=\frac{180(6-2)}{6}=\frac{180(4)}{6}=120$

← Didn't Know|Knew It →

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?

Tap to reveal answer

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.

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

← Didn't Know|Knew It →

0

Knew It