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Intermediate Geometry : Hexagons

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Example Question #16 : How To Find The Length Of The Diagonal Of A Hexagon

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

Possible Answers:

$6\sqrt3$

$3\sqrt3$

$\frac{9\sqrt3}{2}$

$9\sqrt3$

Correct answer:

$9\sqrt3$

Explanation:

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 30-60-90 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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Example Question #17 : How To Find The Length Of The Diagonal Of A Hexagon

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

Possible Answers:

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

$7\sqrt3$

$\frac{5\sqrt3}{3}$

$5\sqrt3$

Correct answer:

$5\sqrt3$

Explanation:

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 30-60-90 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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Example Question #18 : How To Find The Length Of The Diagonal Of A Hexagon

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

Possible Answers:

$12\sqrt3$

$6\sqrt3$

$11\sqrt3$

$\frac{11\sqrt3}{2}$

Correct answer:

$11\sqrt3$

Explanation:

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 30-60-90 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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Example Question #61 : Hexagons

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

Possible Answers:

$\frac{13}{2}$

$\frac{13\sqrt3}{3}$

$13\sqrt3$

Correct answer:

$13\sqrt3$

Explanation:

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 30-60-90 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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Example Question #20 : How To Find The Length Of The Diagonal Of A Hexagon

If the perimeter for the regular hexagon above is 120 , find the length of diagonal .

Possible Answers:

$5\sqrt3$

$10\sqrt3$

$15\sqrt3$

$20\sqrt3$

Correct answer:

$20\sqrt3$

Explanation:

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 30-60-90 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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Example Question #21 : How To Find The Length Of The Diagonal Of A Hexagon

If the perimeter of the regular hexagon above is 168 , find the length of diagonal .

Possible Answers:

$35\sqrt3$

$28\sqrt3$

$14\sqrt3$

$21\sqrt3$

Correct answer:

$28\sqrt3$

Explanation:

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.

$\text{Length of Side}=\frac{168}{6}=28$

Notice that the height of the equilateral triangle creates two congruent 30-60-90 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:

$\frac{28}{2}=\frac{\text{height}}{\sqrt3}$

$\text{height}=14\sqrt3$

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

$\text{Length of Diagonal}=2(14\sqrt3)=28\sqrt3$

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Example Question #22 : How To Find The Length Of The Diagonal Of A Hexagon

If the perimeter of the regular hexagon above is 144 , find the length of diagonal .

Possible Answers:

$24\sqrt3$

$48\sqrt3$

$36\sqrt3$

$12\sqrt3$

Correct answer:

$24\sqrt3$

Explanation:

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.

$\text{Length of Side}=\frac{144}{6}=24$

Notice that the height of the equilateral triangle creates two congruent 30-60-90 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:

$\frac{24}{2}=\frac{\text{height}}{\sqrt3}$

$\text{height}=12\sqrt3$

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

$\text{Length of Diagonal}=2(12\sqrt3)=24\sqrt3$

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Example Question #23 : How To Find The Length Of The Diagonal Of A Hexagon

If the perimeter of the regular hexagon above is 192 , find the length of diagonal .

Possible Answers:

$16\sqrt2$

$32\sqrt3$

$18\sqrt3$

$36\sqrt3$

Correct answer:

$32\sqrt3$

Explanation:

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.

$\text{Length of Side}=\frac{192}{6}=32$

Notice that the height of the equilateral triangle creates two congruent 30-60-90 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:

$\frac{32}{2}=\frac{\text{height}}{\sqrt3}$

$\text{height}=16\sqrt3$

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

$\text{Length of Diagonal}=2(16\sqrt3)=32\sqrt3$

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Example Question #24 : How To Find The Length Of The Diagonal Of A Hexagon

If the perimeter of the regular hexagon above is 276 , what is the length of diagonal ?

Possible Answers:

$39\sqrt3$

$46\sqrt3$

$42\sqrt3$

$23\sqrt3$

Correct answer:

$46\sqrt3$

Explanation:

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.

$\text{Length of Side}=\frac{276}{6}=46$

Notice that the height of the equilateral triangle creates two congruent 30-60-90 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:

$\frac{46}{2}=\frac{\text{height}}{\sqrt3}$

$\text{height}=23\sqrt3$

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

$\text{Length of Diagonal}=2(23\sqrt3)=46\sqrt3$

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Example Question #25 : How To Find The Length Of The Diagonal Of A Hexagon

If the perimeter of the regular hexagon above is 528 , what is the length of diagonal ?

Possible Answers:

$99\sqrt3$

$77\sqrt3$

$88\sqrt3$

$66\sqrt3$

Correct answer:

$88\sqrt3$

Explanation:

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.

$\text{Length of Side}=\frac{528}{6}=88$

Notice that the height of the equilateral triangle creates two congruent 30-60-90 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:

$\frac{88}{2}=\frac{\text{height}}{\sqrt3}$

$\text{height}=44\sqrt3$

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

$\text{Length of Diagonal}=2(44\sqrt3)=88\sqrt3$

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Intermediate Geometry : Hexagons

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #16 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #17 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #18 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #61 : Hexagons

Example Question #20 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #21 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #22 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #23 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #24 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #25 : How To Find The Length Of The Diagonal Of A Hexagon

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