How to find the length of the diagonal of a hexagon - Intermediate Geometry

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Question

A regular hexagon has an apothem of length $7\sqrt{3}$ . Find the area of the circle that encompasses that hexagon:

Answer

When we segment the hexagon into smaller triangles we come out with a nice looking right triangle.

We know our apothem is $7\sqrt{3}$ cm, so that leaves us with a base of 7cm and a hypotenuse of 14 cm.

Looking at the picture, we can see that the hypotenuse of this triangle is also the radius of the outlying circle:

With our radius of 14 cm, we can plug into $Area =\pi r^2$ for circles and come up with an answer of $196\pi$

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Question

Suppose the length of the hexagon has a side length of . What is the diagonal of the hexagon?

Answer

The hexagon is composed of 6 combined equilateral triangles, with 1 vertex from each equilateral joining the center point.

Therefore, since the side length of the hexagon is , and each side length of the equilateral triangle is equal, then all side lengths of the equilateral triangles must be .

Two side lengths of the equilateral triangle joins to create the diagonal of the hexagon.

The diagonal length is .

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Question

Suppose a length of a hexagon is . What must be the diagonal length of the hexagon?

Answer

The hexagon can be broken down into 6 equilateral triangles. Each side of the equilateral triangle is equal. Two of the combined lengths of the equilateral triangles join to form the diagonal of the hexagon.

Therefore, the diagonal is twice the side length of the hexagon.

$20\cdot 2=40$

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Question

If the perimeter of the regular hexagon above is , what is the length of diagonal ?

Answer

When all the diagonals connecting opposite points of a regular hexagon are drawn in, congruent equilateral triangles are created. We can also see that the length of one such diagonal is merely twice the length of a side of the hexagon.

Use the perimeter to find the length of a side of the hexagon.

$\text{Length of Side}=\frac{\text{Perimeter}}{6}$

$\text{Length of Side}=\frac{42}{6}=7$

Double the length of a side to get the length of the wanted diagonal.

$\text{Length of Diagonal}=2(7)=14$

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Question

If the perimeter of the regular hexagon above is , what is the length of diagonal ?

Answer

When all the diagonals connecting opposite points of a regular hexagon are drawn in, congruent equilateral triangles are created. We can also see that the length of one such diagonal is merely twice the length of a side of the hexagon.

Use the perimeter to find the length of a side of the hexagon.

$\text{Length of Side}=\frac{\text{Perimeter}}{6}$

$\text{Length of Side}=\frac{60}{6}=10$

Double the length of a side to get the length of the wanted diagonal.

$\text{Length of Diagonal}=2(10)=20$

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Question

If the perimeter of the regular hexagon above is , what is the length of diagonal ?

Answer

When all the diagonals connecting opposite points of a regular hexagon are drawn in, congruent equilateral triangles are created. We can also see that the length of one such diagonal is merely twice the length of a side of the hexagon.

Use the perimeter to find the length of a side of the hexagon.

$\text{Length of Side}=\frac{\text{Perimeter}}{6}$

$\text{Length of Side}=\frac{66}{6}=11$

Double the length of a side to get the length of the wanted diagonal.

$\text{Length of Diagonal}=2(11)=22$

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Question

If the perimeter of the hexagon above is , what is the length of diagonal ?

Answer

When all the diagonals connecting opposite points of a regular hexagon are drawn in, congruent equilateral triangles are created. We can also see that the length of one such diagonal is merely twice the length of a side of the hexagon.

Use the perimeter to find the length of a side of the hexagon.

$\text{Length of Side}=\frac{\text{Perimeter}}{6}$

$\text{Length of Side}=\frac{72}{6}=12$

Double the length of a side to get the length of the wanted diagonal.

$\text{Length of Diagonal}=2(12)=24$

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Question

If the perimeter of the regular hexagon above is , what is the length of diagonal ?

Answer

When all the diagonals connecting opposite points of a regular hexagon are drawn in, congruent equilateral triangles are created. We can also see that the length of one such diagonal is merely twice the length of a side of the hexagon.

Use the perimeter to find the length of a side of the hexagon.

$\text{Length of Side}=\frac{\text{Perimeter}}{6}$

$\text{Length of Side}=\frac{78}{6}=13$

Double the length of a side to get the length of the wanted diagonal.

$\text{Length of Diagonal}=2(13)=26$

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Question

If the perimeter of the regular hexagon above is , what is the length of diagonal ?

Answer

When all the diagonals connecting opposite points of a regular hexagon are drawn in, congruent equilateral triangles are created. We can also see that the length of one such diagonal is merely twice the length of a side of the hexagon.

Use the perimeter to find the length of a side of the hexagon.

$\text{Length of Side}=\frac{\text{Perimeter}}{6}$

$\text{Length of Side}=\frac{90}{6}=15$

Double the length of a side to get the length of the wanted diagonal.

$\text{Length of Diagonal}=2(15)=30$

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Question

If the perimeter of the regular hexagon above is , what is the length of diagonal ?

Answer

When all the diagonals connecting opposite points of a regular hexagon are drawn in, congruent equilateral triangles are created. We can also see that the length of one such diagonal is merely twice the length of a side of the hexagon.

Use the perimeter to find the length of a side of the hexagon.

$\text{Length of Side}=\frac{\text{Perimeter}}{6}$

$\text{Length of Side}=\frac{33}{6}=5.5$

Double the length of a side to get the length of the wanted diagonal.

$\text{Length of Diagonal}=2(5.5)=11$

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Question

If the perimeter of the regular hexagon above is , what is the length of diagonal ?

Answer

When all the diagonals connecting opposite points of a regular hexagon are drawn in, congruent equilateral triangles are created. We can also see that the length of one such diagonal is merely twice the length of a side of the hexagon.

Use the perimeter to find the length of a side of the hexagon.

$\text{Length of Side}=\frac{\text{Perimeter}}{6}$

$\text{Length of Side}=\frac{45}{6}=7.5$

Double the length of a side to get the length of the wanted diagonal.

$\text{Length of Diagonal}=2(7.5)=15$

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Question

If the perimeter of the regular hexagon above is , what is the length of diagonal ?

Answer

When all the diagonals connecting opposite points of a regular hexagon are drawn in, congruent equilateral triangles are created. We can also see that the length of one such diagonal is merely twice the length of a side of the hexagon.

Use the perimeter to find the length of a side of the hexagon.

$\text{Length of Side}=\frac{\text{Perimeter}}{6}$

$\text{Length of Side}=\frac{105}{6}=17.5$

Double the length of a side to get the length of the wanted diagonal.

$\text{Length of Diagonal}=2(17.5)=35$

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Question

If the perimeter of the regular hexagon above is , what is the length of diagonal ?

Answer

When all the diagonals connecting opposite points of a regular hexagon are drawn in, congruent equilateral triangles are created. We can also see that the length of one such diagonal is merely twice the length of a side of the hexagon.

Use the perimeter to find the length of a side of the hexagon.

$\text{Length of Side}=\frac{\text{Perimeter}}{6}$

$\text{Length of Side}=\frac{231}{6}=38.5$

Double the length of a side to get the length of the wanted diagonal.

$\text{Length of Diagonal}=2(38.5)=77$

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Question

If the perimeter of the regular hexagon is , find the length of diagonal .

Answer

When the regular hexagon is divided into congruent equilateral triangles, it's easy to see that diagonal is comprised of two heights of two equilateral triangles. This holds true for the other diagonals when drawn in, as shown by dotted lines in the figure below:

Now, in order to find the length of the diagonal, we will need to first find the length of a side of the hexagon.

$\text{Length of Side}=\frac{\text{Perimeter}}{6}$

Plug in the given perimeter to find the length of a side for the given hexagon.

$\text{Length of Side}=\frac{12}{6}=2$

Notice that the height of the equilateral triangle creates two congruent triangles whose side lengths are in the ratio of $1:\sqrt3:2$ .

Thus, we can set up the following proportion to find the length of the height:

$\frac{2}{2}=\frac{\text{height}}{\sqrt3}$

$\text{height}=\sqrt3$

Since the diagonal is made up of two of these heights, multiply by to find the length of the diagonal.

$\text{Length of Diagonal}=2\sqrt3$

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Question

If the perimeter of the regular hexagon above is , find the length of diagonal .

Answer

When the regular hexagon is divided into congruent equilateral triangles, it's easy to see that diagonal is comprised of two heights of two equilateral triangles. This holds true for the other diagonals when drawn in, as shown by dotted lines in the figure below:

Now, in order to find the length of the diagonal, we will need to first find the length of a side of the hexagon.

$\text{Length of Side}=\frac{\text{Perimeter}}{6}$

Plug in the given perimeter to find the length of a side for the given hexagon.

$\text{Length of Side}=\frac{42}{6}=7$

Notice that the height of the equilateral triangle creates two congruent triangles whose side lengths are in the ratio of $1:\sqrt3:2$ .

Thus, we can set up the following proportion to find the length of the height:

$\frac{7}{2}=\frac{\text{height}}{\sqrt3}$

$\text{height}=\frac{7\sqrt3}{2}$

Since the diagonal is made up of two of these heights, multiply by to find the length of the diagonal.

$\text{Length of Diagonal}=2(\frac{7\sqrt3}{2})=7\sqrt3$

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Question

If the perimeter of the regular hexagon above is , what is the length of diagonal ?

Answer

When the regular hexagon is divided into congruent equilateral triangles, it's easy to see that diagonal is comprised of two heights of two equilateral triangles. This holds true for the other diagonals when drawn in, as shown by dotted lines in the figure below:

Now, in order to find the length of the diagonal, we will need to first find the length of a side of the hexagon.

$\text{Length of Side}=\frac{\text{Perimeter}}{6}$

Plug in the given perimeter to find the length of a side for the given hexagon.

$\text{Length of Side}=\frac{54}{6}=9$

Notice that the height of the equilateral triangle creates two congruent triangles whose side lengths are in the ratio of $1:\sqrt3:2$ .

Thus, we can set up the following proportion to find the length of the height:

$\frac{9}{2}=\frac{\text{height}}{\sqrt3}$

$\text{height}=\frac{9\sqrt3}{2}$

Since the diagonal is made up of two of these heights, multiply by to find the length of the diagonal.

$\text{Length of Diagonal}=2(\frac{9\sqrt3}{2})=9\sqrt3$

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Question

If the perimeter of the regular hexagon is , find the length of diagonal .

Answer

When the regular hexagon is divided into congruent equilateral triangles, it's easy to see that diagonal is comprised of two heights of two equilateral triangles. This holds true for the other diagonals when drawn in, as shown by dotted lines in the figure below:

Now, in order to find the length of the diagonal, we will need to first find the length of a side of the hexagon.

$\text{Length of Side}=\frac{\text{Perimeter}}{6}$

Plug in the given perimeter to find the length of a side for the given hexagon.

$\text{Length of Side}=\frac{66}{6}=11$

Notice that the height of the equilateral triangle creates two congruent triangles whose side lengths are in the ratio of $1:\sqrt3:2$ .

Thus, we can set up the following proportion to find the length of the height:

$\frac{11}{2}=\frac{\text{height}}{\sqrt3}$

$\text{height}=\frac{11\sqrt3}{2}$

Since the diagonal is made up of two of these heights, multiply by to find the length of the diagonal.

$\text{Length of Diagonal}=2(\frac{11\sqrt3}{2})=11\sqrt3$

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Question

If the perimeter of the regular hexagon above is , find the length of diagonal .

Answer

When the regular hexagon is divided into congruent equilateral triangles, it's easy to see that diagonal is comprised of two heights of two equilateral triangles. This holds true for the other diagonals when drawn in, as shown by dotted lines in the figure below:

Now, in order to find the length of the diagonal, we will need to first find the length of a side of the hexagon.

$\text{Length of Side}=\frac{\text{Perimeter}}{6}$

Plug in the given perimeter to find the length of a side for the given hexagon.

$\text{Length of Side}=\frac{30}{6}=5$

Notice that the height of the equilateral triangle creates two congruent triangles whose side lengths are in the ratio of $1:\sqrt3:2$ .

Thus, we can set up the following proportion to find the length of the height:

$\frac{5}{2}=\frac{\text{height}}{\sqrt3}$

$\text{height}=\frac{5\sqrt3}{2}$

Since the diagonal is made up of two of these heights, multiply by to find the length of the diagonal.

$\text{Length of Diagonal}=2(\frac{5\sqrt3}{2})=5\sqrt3$

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Question

If the perimeter of the regular hexagon above is , what is the length of diagonal ?

Answer

When the regular hexagon is divided into congruent equilateral triangles, it's easy to see that diagonal is comprised of two heights of two equilateral triangles. This holds true for the other diagonals when drawn in, as shown by dotted lines in the figure below:

Now, in order to find the length of the diagonal, we will need to first find the length of a side of the hexagon.

$\text{Length of Side}=\frac{\text{Perimeter}}{6}$

Plug in the given perimeter to find the length of a side for the given hexagon.

$\text{Length of Side}=\frac{78}{6}=13$

Notice that the height of the equilateral triangle creates two congruent triangles whose side lengths are in the ratio of $1:\sqrt3:2$ .

Thus, we can set up the following proportion to find the length of the height:

$\frac{13}{2}=\frac{\text{height}}{\sqrt3}$

$\text{height}=\frac{13\sqrt3}{2}$

Since the diagonal is made up of two of these heights, multiply by to find the length of the diagonal.

$\text{Length of Diagonal}=2(\frac{13\sqrt3}{2})=13\sqrt3$

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Question

If the perimeter for the regular hexagon above is , find the length of diagonal .

Answer

When the regular hexagon is divided into congruent equilateral triangles, it's easy to see that diagonal is comprised of two heights of two equilateral triangles. This holds true for the other diagonals when drawn in, as shown by dotted lines in the figure below:

Now, in order to find the length of the diagonal, we will need to first find the length of a side of the hexagon.

$\text{Length of Side}=\frac{\text{Perimeter}}{6}$

Plug in the given perimeter to find the length of a side for the given hexagon.

$\text{Length of Side}=\frac{120}{6}=20$

Notice that the height of the equilateral triangle creates two congruent triangles whose side lengths are in the ratio of $1:\sqrt3:2$ .

Thus, we can set up the following proportion to find the length of the height:

$\frac{20}{2}=\frac{\text{height}}{\sqrt3}$

$\text{height}=10\sqrt3$

Since the diagonal is made up of two of these heights, multiply by to find the length of the diagonal.

$\text{Length of Diagonal}=2(10\sqrt3)=20\sqrt3$

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