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Intermediate Geometry : Hexagons

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Example Question #31 : How To Find The Perimeter Of A Hexagon

If the area of a regular hexagon is $100\sqrt2$ , what is the perimeter of the hexagon?

Possible Answers:

49.55

42.01

44.27

41.68

Correct answer:

44.27

Explanation:

Recall the formula to find the area of a regular hexagon:

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

Rearrange the formula to solve for the side.

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

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

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

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

Substitute in for the side to solve for the perimeter.

$\text{Perimeter}=6\sqrt{\frac{2(\text{Area})}{3\sqrt3}}$

Plug in the given area and use a calculator to find the perimeter.

$\text{Perimeter}=6\sqrt{\frac{2(100\sqrt2)}{3\sqrt3}}=44.27$

Make sure to round to the two places after the decimal.

Report an Error

Example Question #32 : How To Find The Perimeter Of A Hexagon

If the area of a regular hexagon is $150\sqrt5$ , what is the perimeter of the hexagon?

Possible Answers:

75.82

68.17

69.20

73.72

Correct answer:

68.17

Explanation:

Recall the formula to find the area of a regular hexagon:

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

Rearrange the formula to solve for the side.

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

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

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

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

Substitute in for the side to solve for the perimeter.

$\text{Perimeter}=6\sqrt{\frac{2(\text{Area})}{3\sqrt3}}$

Plug in the given area and use a calculator to find the perimeter.

$\text{Perimeter}=6\sqrt{\frac{2(150\sqrt5)}{3\sqrt3}}=68.17$

Make sure to round to the two places after the decimal.

Report an Error

Example Question #33 : How To Find The Perimeter Of A Hexagon

If the area of a regular hexagon is $\frac{4\sqrt5}{3}$ , what is the perimeter of the hexagon?

Possible Answers:

5.10

6.43

8.05

4.19

Correct answer:

6.43

Explanation:

Recall the formula to find the area of a regular hexagon:

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

Rearrange the formula to solve for the side.

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

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

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

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

Substitute in for the side to solve for the perimeter.

$\text{Perimeter}=6\sqrt{\frac{2(\text{Area})}{3\sqrt3}}$

Plug in the given area and use a calculator to find the perimeter.

$\text{Perimeter}=6\sqrt{\frac{2(\frac{4\sqrt5}{3})}{3\sqrt3}}=6.43$

Make sure to round to the two places after the decimal.

Report an Error

Example Question #34 : How To Find The Perimeter Of A Hexagon

Find the perimeter of the regular hexagon below.

Possible Answers:

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

$54\sqrt3$

$48\sqrt3$

$\frac{101\sqrt3}{3}$

Correct answer:

$48\sqrt3$

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:

$\frac{12}{\sqrt3}=\frac{\text{base}}{1}$

The length of the base is then $\frac{12}{\sqrt3}=4\sqrt3$ .

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:

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

$\text{side}=2(4\sqrt3)=8\sqrt3$

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:

$\text{Perimeter}=6(8\sqrt3)=48\sqrt3$

Report an Error

Example Question #35 : How To Find The Perimeter Of A Hexagon

Find the perimeter of the regular hexagon below:

Possible Answers:

$72\sqrt3$

$60\sqrt3$

$56\sqrt3$

$64\sqrt3$

Correct answer:

$56\sqrt3$

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 $\frac{14}{\sqrt3}=\frac{14\sqrt3}{3}$ .

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:

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

$\text{side}=2(\frac{14\sqrt3}{3})=\frac{28\sqrt3}{3}$

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:

$\text{Perimeter}=6(\frac{28\sqrt3}{3})=56\sqrt3$

Report an Error

Example Question #36 : How To Find The Perimeter Of A Hexagon

Find the perimeter of the regular hexagon below:

Possible Answers:

$21\sqrt3$

$35\sqrt3$

$14\sqrt3$

$28\sqrt3$

Correct answer:

$28\sqrt3$

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:

$\frac{7}{\sqrt3}=\frac{\text{base}}{1}$

The length of the base is then $\frac{7}{\sqrt3}=\frac{7\sqrt3}{3}$ .

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:

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

$\text{side}=2(\frac{7\sqrt3}{3})=\frac{14\sqrt3}{3}$

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:

$\text{Perimeter}=6(\frac{14\sqrt3}{3})=28\sqrt3$

Report an Error

Example Question #31 : How To Find The Perimeter Of A Hexagon

Find the perimeter of the regular hexagon below:

Possible Answers:

$18\sqrt3$

$12\sqrt3$

$15\sqrt3$

$9\sqrt3$

Correct answer:

$12\sqrt3$

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:

$\frac{3}{\sqrt3}=\frac{\text{base}}{1}$

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

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:

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

$\text{side}=2\sqrt3$

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:

$\text{Perimeter}=6(2\sqrt3)=12\sqrt3$

Report an Error

Example Question #38 : How To Find The Perimeter Of A Hexagon

Find the perimeter of the regular hexagon below:

Possible Answers:

$108\sqrt3$

$96\sqrt3$

$114\sqrt3$

$102\sqrt3$

Correct answer:

$96\sqrt3$

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:

$\frac{24}{\sqrt3}=\frac{\text{base}}{1}$

The length of the base is then $\frac{24}{\sqrt3}=8\sqrt3$ .

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:

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

$\text{side}=2(8\sqrt3)=16\sqrt3$

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:

$\text{Perimeter}=6(16\sqrt3)=96\sqrt3$

Report an Error

Example Question #39 : How To Find The Perimeter Of A Hexagon

Find the perimeter of the regular hexagon below:

Possible Answers:

$\frac{40\sqrt3}{3}$

$40\sqrt3$

$30\sqrt3$

$50\sqrt3$

Correct answer:

$40\sqrt3$

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:

$\frac{10}{\sqrt3}=\frac{\text{base}}{1}$

The length of the base is then $\frac{10}{\sqrt3}=\frac{10\sqrt3}{3}$ .

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:

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

$\text{side}=2(\frac{10\sqrt3}{3})=\frac{20\sqrt3}{3}$

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:

$\text{Perimeter}=6(\frac{20\sqrt3}{3})=40\sqrt3$

Report an Error

Example Question #40 : How To Find The Perimeter Of A Hexagon

Find the perimeter of the regular hexagon below:

Possible Answers:

$108\sqrt3$

$104\sqrt3$

$112\sqrt3$

$96\sqrt3$

Correct answer:

$104\sqrt3$

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 $\frac{26}{\sqrt3}=\frac{26\sqrt3}{3}$ .

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:

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

$\text{side}=2(\frac{26\sqrt3}{3})=\frac{52\sqrt3}{3}$

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:

$\text{Perimeter}=6(\frac{52\sqrt3}{3})=104\sqrt3$

Report an Error

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Intermediate Geometry : Hexagons

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #31 : How To Find The Perimeter Of A Hexagon

Example Question #32 : How To Find The Perimeter Of A Hexagon

Example Question #33 : How To Find The Perimeter Of A Hexagon

Example Question #34 : How To Find The Perimeter Of A Hexagon

Example Question #35 : How To Find The Perimeter Of A Hexagon

Example Question #36 : How To Find The Perimeter Of A Hexagon

Example Question #31 : How To Find The Perimeter Of A Hexagon

Example Question #38 : How To Find The Perimeter Of A Hexagon

Example Question #39 : How To Find The Perimeter Of A Hexagon

Example Question #40 : How To Find The Perimeter Of A Hexagon

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