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Intermediate Geometry : Hexagons

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

Find the length of a side of a regular hexagon that has its center at .

Possible Answers:

$16\sqrt3$

$8\sqrt3$

$24\sqrt3$

$30\sqrt3$

Correct answer:

$16\sqrt3$

Explanation:

This is a regular hexagon; therefore, when we draw in the diagonals, we will create identical equilateral triangles as shown by the figure:

Now, the given line segment also serves as the height to the newly created equilateral triangles. This height also splits an equilateral triangle into two congruent $30^\circ-60^\circ-90^\circ$ triangles.

Recall that $30^\circ-60^\circ-90^\circ$ triangles have side lengths that are in the ratio of $1:\sqrt3:2$ .

Substitute in the given height into the ratio in order to find the length of the base of the $30^\circ-60^\circ-90^\circ$ triangle.

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

Substitute.

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

Solve.

$\text{base}=\frac{24\sqrt3}{3}=8\sqrt3$

The length of the side of the hexagon is twice the length of the base.

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

Substitute in the value of the length of the base to find the side length of the hexagon.

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

Solve.

$\text{side length}=16\sqrt3$

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

Find the length of a side of a regular hexagon that has its center at .

Possible Answers:

$18\sqrt3$

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

$\frac{52\sqrt3}{3}$

$\frac{55\sqrt3}{3}$

Correct answer:

$\frac{52\sqrt3}{3}$

Explanation:

This is a regular hexagon; therefore, when we draw in the diagonals, we will create identical equilateral triangles as shown by the figure:

Recall that $30^\circ-60^\circ-90^\circ$ triangles have side lengths that are in the ratio of $1:\sqrt3:2$ .

Substitute in the given height into the ratio in order to find the length of the base of the $30^\circ-60^\circ-90^\circ$ triangle.

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

Substitute.

$\frac{26}{\sqrt3}=\frac{\text{base}}{1}$

Solve.

$\text{base}=\frac{26\sqrt3}{3}$

The length of the side of the hexagon is twice the length of the base.

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

Substitute in the value of the length of the base to find the side length of the hexagon.

$\text{side length}=2\left(\frac{26\sqrt3}{3}\right)$

Solve.

$\text{side length}=\frac{52\sqrt3}{3}$

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

Find the length of a side of a regular hexagon that has its center at .

Possible Answers:

$19\sqrt3$

$30\sqrt3$

$20\sqrt3$

$10\sqrt3$

Correct answer:

$20\sqrt3$

Explanation:

This is a regular hexagon; therefore, when we draw in the diagonals, we will create identical equilateral triangles as shown by the figure:

Recall that $30^\circ-60^\circ-90^\circ$ triangles have side lengths that are in the ratio of $1:\sqrt3:2$ .

Substitute in the given height into the ratio in order to find the length of the base of the $30^\circ-60^\circ-90^\circ$ triangle.

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

Substitute.

$\frac{30}{\sqrt3}=\frac{\text{base}}{1}$

Solve.

$\text{base}=\frac{30\sqrt3}{3}=10\sqrt3$

The length of the side of the hexagon is twice the length of the base.

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

Substitute in the value of the length of the base to find the side length of the hexagon.

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

Solve.

$\text{side length}=20\sqrt3$

Report an Error

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

Find the length of a side of a regular hexagon that has its center at .

Possible Answers:

$\frac{32\sqrt3}{3}$

$\frac{125\sqrt3}{3}$

$\frac{128\sqrt3}{3}$

$\frac{64\sqrt3}{3}$

Correct answer:

$\frac{64\sqrt3}{3}$

Explanation:

This is a regular hexagon; therefore, when we draw in the diagonals, we will create identical equilateral triangles as shown by the figure:

Recall that $30^\circ-60^\circ-90^\circ$ triangles have side lengths that are in the ratio of $1:\sqrt3:2$ .

Substitute in the given height into the ratio in order to find the length of the base of the $30^\circ-60^\circ-90^\circ$ triangle.

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

Substitute.

$\frac{32}{\sqrt3}=\frac{\text{base}}{1}$

Solve.

$\text{base}=\frac{32\sqrt3}{3}$

The length of the side of the hexagon is twice the length of the base.

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

Substitute in the value of the length of the base to find the side length of the hexagon.

$\text{side length}=2\left(\frac{32\sqrt3}{3}\right)$

Solve.

$\text{side length}=\frac{64\sqrt3}{3}$

Report an Error

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

Find the length of a side of a regular hexagon that has its center at .

Possible Answers:

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

$\frac{10\sqrt3}{3}$

$4\sqrt3$

$5\sqrt3$

Correct answer:

$\frac{10\sqrt3}{3}$

Explanation:

This is a regular hexagon; therefore, when we draw in the diagonals, we will create identical equilateral triangles as shown by the figure:

Recall that $30^\circ-60^\circ-90^\circ$ triangles have side lengths that are in the ratio of $1:\sqrt3:2$ .

Substitute in the given height into the ratio in order to find the length of the base of the $30^\circ-60^\circ-90^\circ$ triangle.

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

Substitute.

$\frac{5}{\sqrt3}=\frac{\text{base}}{1}$

Solve.

$\text{base}=\frac{5\sqrt3}{3}$

The length of the side of the hexagon is twice the length of the base.

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

Substitute in the value of the length of the base to find the side length of the hexagon.

$\text{side length}=2\left(\frac{5\sqrt3}{3}\right)$

Solve.

$\text{side length}=\frac{10\sqrt3}{3}$

Report an Error

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

Find the length of a side of a regular hexagon that has its center at .

Possible Answers:

$4\sqrt3$

$\frac{19\sqrt3}{3}$

$\frac{10\sqrt3}{3}$

$\frac{16\sqrt3}{3}$

Correct answer:

$\frac{16\sqrt3}{3}$

Explanation:

This is a regular hexagon; therefore, when we draw in the diagonals, we will create identical equilateral triangles as shown by the figure:

Recall that $30^\circ-60^\circ-90^\circ$ triangles have side lengths that are in the ratio of $1:\sqrt3:2$ .

Substitute in the given height into the ratio in order to find the length of the base of the $30^\circ-60^\circ-90^\circ$ triangle.

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

Substitute.

$\frac{8}{\sqrt3}=\frac{\text{base}}{1}$

Solve.

$\text{base}=\frac{8\sqrt3}{3}$

The length of the side of the hexagon is twice the length of the base.

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

Substitute in the value of the length of the base to find the side length of the hexagon.

$\text{side length}=2\left(\frac{8\sqrt3}{3}\right)$

Solve.

$\text{side length}=\frac{16\sqrt3}{3}$

Report an Error

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

Find the length of a side of a regular hexagon that has its center at .

Possible Answers:

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

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

$\frac{4\sqrt3}{3}$

$\frac{7\sqrt3}{3}$

Correct answer:

$\frac{4\sqrt3}{3}$

Explanation:

This is a regular hexagon; therefore, when we draw in the diagonals, we will create identical equilateral triangles as shown by the figure:

Recall that $30^\circ-60^\circ-90^\circ$ triangles have side lengths that are in the ratio of $1:\sqrt3:2$ .

Substitute in the given height into the ratio in order to find the length of the base of the $30^\circ-60^\circ-90^\circ$ triangle.

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

Substitute.

$\frac{2}{\sqrt3}=\frac{\text{base}}{1}$

Solve.

$\text{base}=\frac{2\sqrt3}{3}$

The length of the side of the hexagon is twice the length of the base.

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

Substitute in the value of the length of the base to find the side length of the hexagon.

$\text{side length}=2\left(\frac{2\sqrt3}{3}\right)$

Solve.

$\text{side length}=\frac{4\sqrt3}{3}$

Report an Error

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

Given: Regular Hexagon HEXAGO with center . Construct segments $\overline{CA}$ and $\overline{CO}$ to form Quadrilateral CAGO .

True or false: Quadrilateral CAGO is a rhombus.

Possible Answers:

False

True

Correct answer:

True

Explanation:

Below is regular Hexagon HEXAGO with center , a segment drawn from to each vertex - that is, each of its radii drawn.

Hexagon 2

Each angle of a regular hexagon measures $120^{\circ }$ ; by symmetry, each radius bisects an angle of the hexagon, so

$m \angle CAG = m \angle CGA = m\angle CGO = m \angle COG = 60 ^{\circ }$ .

Also, each central angle is one sixth of a complete circle, so $m \angle OCG = m \angle GCA = 60 ^{\circ }$ .

This makes $\bigtriangleup OCG$ and $\bigtriangleup GCA$ equilateral triangles. Therefore,

CO = OG = CG

and

CG = GA = AC

By transitivity,

CO = OG = GA = AC .

Quadrilateral CAGO has four sides of equal length and is consequently a rhombus.

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Example Question #81 : Hexagons

Five sides of a hexagon are known to have length 10; the perimeter of the pentagon is 60.

True, false, or undetermined: The hexagon is regular.

Possible Answers:

False

True

Undetermined

Correct answer:

Undetermined

Explanation:

If the hexagon has perimeter 60, and five of its sides are known to have length 10, then the length of its sixth side is

60 - (10+ 10+ 10 + 10 + 10 ) = 60-50 = 10 .

All six sides are of the same length, so the hexagon is equilateral. However, for the hexagon to be regular, it must hold that all six angles are of the same measure as well - that is, it must also be equiangular. However, an equilateral hexagon may or may not be equiangular, as seen by the figures below; both are equilateral hexagons, but only the left one is equiangular.

Hexagons

Therefore, the question of whether the hexagon is regular cannot be answered without further information.

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Example Question #1 : How To Find An Angle In A Hexagon

There is a regular hexagon with a side length of $4 \:cm$ . What is the measure of an internal angle?

Possible Answers:

$180^{\circ}$

cannot be determined

$60^{\circ}$

$720^{\circ}$

$120^{\circ}$

Correct answer:

$120^{\circ}$

Explanation:

Given that the hexagon is a regular hexagon, this means that all the side length are congruent and all internal angles are congruent. The question requires us to solve for the measure of an internal angle. Given the aforementioned definition of a regular polygon, this means that there must only be one correct answer.

In order to solve for the answer, the question provides additional information that isn't necessarily required. The measure of an internal angle can be solved for using the equation:

$\frac{180^{\circ}(n-2)}{n}$ where is the number of sides of the polygon.

In this case, n = 6 .

For this problem, the information about the side length may be negated.

$\frac{180^{\circ}(6-2)}{6}$

$\frac{180^{\circ}(4)}{6}$

$\frac{180^{\circ}(2)}{3}$

$120^{\circ}$

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Intermediate Geometry : Hexagons

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

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

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

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

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

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

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

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

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

Example Question #81 : Hexagons

Example Question #1 : How To Find An Angle In A Hexagon

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