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Intermediate Geometry : How to find the perimeter of a hexagon

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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 #42 : How To Find The Perimeter Of A Hexagon

Find the perimeter of the regular hexagon below:

Possible Answers:

$12\sqrt3$

$8\sqrt3$

$10\sqrt3$

$14\sqrt3$

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:

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

Report an Error

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

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Intermediate Geometry : How to find the perimeter of a hexagon

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 #42 : How To Find The Perimeter Of A Hexagon

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

All Intermediate Geometry Resources

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