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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #1 : 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:

$11\sqrt3$

$44\sqrt3$

Correct answer:

Explanation:

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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Example Question #1 : How To Find The Length Of The Diagonal Of A Hexagon

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

Possible Answers:

Correct answer:

Explanation:

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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Example Question #1 : 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:

$12\sqrt3$

$14\sqrt3$

Correct answer:

Explanation:

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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Example Question #9 : 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:

$20\sqrt3$

$15\sqrt3$

Correct answer:

Explanation:

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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Example Question #10 : 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:

7.5

5.5

Correct answer:

Explanation:

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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Example Question #11 : 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:

$9\sqrt3$

Correct answer:

Explanation:

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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Example Question #12 : How To Find The Length Of The Diagonal Of A Hexagon

If the perimeter of the regular hexagon above is 105 , what is the length of diagonal ?

Possible Answers:

$30\sqrt3$

Correct answer:

Explanation:

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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Example Question #13 : How To Find The Length Of The Diagonal Of A Hexagon

If the perimeter of the regular hexagon above is 231 , what is the length of diagonal ?

Possible Answers:

Correct answer:

Explanation:

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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Example Question #14 : 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:

$\sqrt3$

The length of the diagonal cannot be determined.

$2\sqrt3$

Correct answer:

$2\sqrt3$

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.

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

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:

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

$\text{height}=\sqrt3$

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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Example Question #15 : 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:

$7\sqrt3$

$\frac{3\sqrt3}{2}$

$14\sqrt3$

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

Correct answer:

$7\sqrt3$

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.

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

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:

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

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

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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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #1 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #1 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #1 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #9 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #10 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #11 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #12 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #13 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #14 : How To Find The Length Of The Diagonal Of A Hexagon

Example Question #15 : How To Find The Length Of The Diagonal Of A Hexagon

All Intermediate Geometry Resources

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