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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #41 : How To Find The Perimeter Of A Hexagon

Find the perimeter of the regular hexagon below:

Possible Answers:

$150\sqrt3$

$120\sqrt3$

$90\sqrt3$

$180\sqrt3$

Correct answer:

$120\sqrt3$

Explanation:

When all the diagonals of the regular hexagon are drawn in, you should notice that six congruent equilateral triangles are created.

Thus, we know that the given line segment, also known as an apothem, acts as the height of an equilateral triangle. The apothem also creates two congruent 30-60-90 triangles.

Thus, we can use the ratio of the side lengths to find the length of the base of one of the 30-60-90 triangles.

Recall that the ratios of the side lengths in a 30-60-90 triangle are $1:\sqrt3:2$ .

We can then set up the following equation:

$\frac{14}{\sqrt3}=\frac{\text{base}}{1}$

The length of the base is then $\frac{30}{\sqrt3}=10\sqrt3$ .

Now, the length of this base is half the length of a side of the hexagon. Multiply the base by two to find the length of a side:

$\text{side}=2(\text{base})$

$\text{side}=2(10\sqrt3)=20\sqrt3$

Next, recall how to find the perimeter of a regular hexagon:

$\text{Perimeter}=6(\text{side})$

Plug in the side length to find the perimeter:

$\text{Perimeter}=6(20\sqrt3)=120\sqrt3$

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Example Question #41 : Hexagons

Find the perimeter of the regular hexagon below:

Possible Answers:

$12\sqrt3$

$14\sqrt3$

$8\sqrt3$

$10\sqrt3$

Correct answer:

$8\sqrt3$

Explanation:

When all the diagonals of the regular hexagon are drawn in, you should notice that six congruent equilateral triangles are created.

Thus, we know that the given line segment, also known as an apothem, acts as the height of an equilateral triangle. The apothem also creates two congruent 30-60-90 triangles.

Thus, we can use the ratio of the side lengths to find the length of the base of one of the 30-60-90 triangles.

Recall that the ratios of the side lengths in a 30-60-90 triangle are $1:\sqrt3:2$ .

We can then set up the following equation:

$\frac{14}{\sqrt3}=\frac{\text{base}}{1}$

The length of the base is then $\frac{2}{\sqrt3}=\frac{2\sqrt3}{3}$ .

Now, the length of this base is half the length of a side of the hexagon. Multiply the base by two to find the length of a side:

$\text{side}=2(\text{base})$

$\text{side}=2(\frac{2\sqrt3}{3})=\frac{4\sqrt3}{3}$

Next, recall how to find the perimeter of a regular hexagon:

$\text{Perimeter}=6(\text{side})$

Plug in the side length to find the perimeter:

$\text{Perimeter}=6(\frac{4\sqrt3}{3})=8\sqrt3$

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Example Question #43 : How To Find The Perimeter Of A Hexagon

Find the perimeter of the regular hexagon below:

Possible Answers:

$109\sqrt3$

$108\sqrt3$

$100\sqrt3$

$116\sqrt3$

Correct answer:

$108\sqrt3$

Explanation:

When all the diagonals of the regular hexagon are drawn in, you should notice that six congruent equilateral triangles are created.

Thus, we know that the given line segment, also known as an apothem, acts as the height of an equilateral triangle. The apothem also creates two congruent 30-60-90 triangles.

Thus, we can use the ratio of the side lengths to find the length of the base of one of the 30-60-90 triangles.

Recall that the ratios of the side lengths in a 30-60-90 triangle are $1:\sqrt3:2$ .

We can then set up the following equation:

$\frac{14}{\sqrt3}=\frac{\text{base}}{1}$

The length of the base is then $\frac{27}{\sqrt3}=9\sqrt3$ .

Now, the length of this base is half the length of a side of the hexagon. Multiply the base by two to find the length of a side:

$\text{side}=2(\text{base})$

$\text{side}=2(9\sqrt3)=18\sqrt3$

Next, recall how to find the perimeter of a regular hexagon:

$\text{Perimeter}=6(\text{side})$

Plug in the side length to find the perimeter:

$\text{Perimeter}=6(18\sqrt3)=108\sqrt3$

Report an Error

Example Question #44 : How To Find The Perimeter Of A Hexagon

Find the perimeter of the regular hexagon below:

Possible Answers:

$96\sqrt3$

$160\sqrt3$

$128\sqrt3$

$136\sqrt3$

Correct answer:

$128\sqrt3$

Explanation:

When all the diagonals of the regular hexagon are drawn in, you should notice that six congruent equilateral triangles are created.

Thus, we know that the given line segment, also known as an apothem, acts as the height of an equilateral triangle. The apothem also creates two congruent 30-60-90 triangles.

Thus, we can use the ratio of the side lengths to find the length of the base of one of the 30-60-90 triangles.

Recall that the ratios of the side lengths in a 30-60-90 triangle are $1:\sqrt3:2$ .

We can then set up the following equation:

$\frac{14}{\sqrt3}=\frac{\text{base}}{1}$

The length of the base is then $\frac{32}{\sqrt3}=\frac{32\sqrt3}{3}$ .

Now, the length of this base is half the length of a side of the hexagon. Multiply the base by two to find the length of a side:

$\text{side}=2(\text{base})$

$\text{side}=2(\frac{32\sqrt3}{3})=\frac{64\sqrt3}{3}$

Next, recall how to find the perimeter of a regular hexagon:

$\text{Perimeter}=6(\text{side})$

Plug in the side length to find the perimeter:

$\text{Perimeter}=6(\frac{64\sqrt3}{3})=128\sqrt3$

Report an Error

Example Question #45 : How To Find The Perimeter Of A Hexagon

Find the perimeter of the regular hexagon below:

Possible Answers:

$205\sqrt3$

$220\sqrt3$

$135\sqrt3$

$180\sqrt3$

Correct answer:

$180\sqrt3$

Explanation:

When all the diagonals of the regular hexagon are drawn in, you should notice that six congruent equilateral triangles are created.

Thus, we know that the given line segment, also known as an apothem, acts as the height of an equilateral triangle. The apothem also creates two congruent 30-60-90 triangles.

Thus, we can use the ratio of the side lengths to find the length of the base of one of the 30-60-90 triangles.

Recall that the ratios of the side lengths in a 30-60-90 triangle are $1:\sqrt3:2$ .

We can then set up the following equation:

$\frac{14}{\sqrt3}=\frac{\text{base}}{1}$

The length of the base is then $\frac{45}{\sqrt3}=15\sqrt3$ .

Now, the length of this base is half the length of a side of the hexagon. Multiply the base by two to find the length of a side:

$\text{side}=2(\text{base})$

$\text{side}=2(15\sqrt3)=30\sqrt3$

Next, recall how to find the perimeter of a regular hexagon:

$\text{Perimeter}=6(\text{side})$

Plug in the side length to find the perimeter:

$\text{Perimeter}=6(30\sqrt3)=180\sqrt3$

Report an Error

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

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

Possible Answers:

None of the other answers

$212\pi$ cm^2

$196\pi$ cm^2

$144\pi$ cm^2

$108\pi$ cm^2

Correct answer:

$196\pi$ cm^2

Explanation:

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 $196\pi$ cm^2

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Example Question #2 : How To Find The Length Of The Diagonal Of A Hexagon

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

Possible Answers:

$\frac{1}{6}a+\frac{1}{6}b$

6a+6b

a+b

3a+3b

2a+2b

Correct answer:

2a+2b

Explanation:

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 a+b , and each side length of the equilateral triangle is equal, then all side lengths of the equilateral triangles must be a+b .

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

2(a+b)=2a+2b

The diagonal length is 2a+2b .

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

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

Possible Answers:

Correct answer:

Explanation:

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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Example Question #4 : 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:

$7\sqrt3$

$\frac{7\sqrt3}{2}$

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.

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

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

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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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:

$10\sqrt3$

$20\sqrt3$

Correct answer:

Explanation:

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

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

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

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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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #41 : How To Find The Perimeter Of A Hexagon

Example Question #41 : Hexagons

Example Question #43 : How To Find The Perimeter Of A Hexagon

Example Question #44 : How To Find The Perimeter Of A Hexagon

Example Question #45 : How To Find The Perimeter Of A Hexagon

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

Example Question #2 : 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 #4 : 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

All Intermediate Geometry Resources

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