How to find the length of the side of a hexagon - Intermediate Geometry

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Question

An apothem is a line drawn from the center of a regular shape to the center of one of its edges. The line drawn is perpendicular to the edge. A regular hexagon has an apothem of $6\sqrt3:cm$ . Find the length of one of the sides.

Answer

If the apothem is $6\sqrt3:cm$ and the question requires us to solve for the length of one of the sides, the problem can be resolved through the use of right triangles and trig functions.

As long as one angle and one side length is known for a right triangle, trig functions can be used to solve for a mystery side. In the previous image, the side of interest has been labeled as . Keep in mind that is actually half the length of one side of the regular hexagon. In order to solve for , the first step is to solve for the measure of an internal angle of the hexagon. This angle has been marked in the image. This can be solved for by using:

$\frac{180^{\circ}(n-2)}{n}$ where is the number of sides of the hexagon. In this problem, .

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

$\frac{180^{\circ}(4)}{6} = 120^{\circ}$

This measure of $120^{\circ}$ is the measure of the entire angle. Keep in mind that the drawn triangle is actually bisecting the internal angle. This means that the angle of interest in the triangle is actually $60^{\circ}$ .

Now that we have the apothem and one of the angles, we can use trig functions to solve for .

Using SOH CAH TOA, we realize that this problem would require us to use the tan function where the ratio would be $\frac{opp}{adj}$ , or for this problem's data, $\frac{6\sqrt3:cm}{x}$ .

$tan(60^\circ)=\frac{6\sqrt{3}:cm}{x}$

$x \cdot tan(60^\circ)= 6\sqrt{3}:cm$

$x= \frac{6\sqrt{3}:cm}{tan(60^\circ)}$

Now must be multiplied by because it's the length of half of the total length of one side of the hexagon. Therefore, the final answer is .

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Question

Suppose the perimeter of a hexagon is . What is the side length of the hexagon?

Answer

Write the perimeter formula for hexagons.

Substitute the perimeter in the formula and solve for the side length.

$s=4a+\frac{15}{6}b=4a+\frac{5}{2}b$

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Question

Let the perimeter of a hexagon be . What is a side length of the hexagon?

Answer

Write the formula for finding the perimeter of a hexagon.

Substitute the perimeter and solve for the side length.

$s=\frac{15}{6}=\frac{5}{2}$

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Question

If the perimeter of a regular hexagon is , what must be the side length of the hexagon?

Answer

Write the formula for the perimeter of a hexagon.

Substitute the perimeter and solve for the side.

$s=\frac{15}{6}a+\frac{16}{6}b= \frac{5}{2}a+\frac{8}{3}b$

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Question

If the perimeter of a hexagon is $3+6\sqrt3$ , what is the side length?

Answer

Write the perimeter formula for a hexagon.

Substitute the perimeter into the formula and solve for the side length.

$3+6\sqrt3=6s$

$s=\frac{3+6\sqrt3}{6}= \frac{1+2\sqrt3}{2}$

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Question

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

Answer

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.

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

Solve.

$\text{base}=\frac{12\sqrt3}{3}=4\sqrt3$

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.

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

Solve.

$\text{side length}=8\sqrt3$

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Question

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

Answer

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.

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

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.

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

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Question

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

Answer

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.

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

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.

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

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Question

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

Answer

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.

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

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.

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

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Question

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

Answer

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.

$\frac{16}{\sqrt3}=\frac{\text{base}}{1}$

Solve.

$\text{base}=\frac{16\sqrt3}{3}$

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.

$\text{side length}=2\left(\frac{16\sqrt3}{3}\right)$

Solve.

$\text{side length}=\frac{32\sqrt3}{3}$

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Question

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

Answer

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.

$\frac{22}{\sqrt3}=\frac{\text{base}}{1}$

Solve.

$\text{base}=\frac{22\sqrt3}{3}$

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.

$\text{side length}=2\left(\frac{22\sqrt3}{3}\right)$

Solve.

$\text{side length}=\frac{44\sqrt3}{3}$

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Question

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

Answer

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.

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

Solve.

$\text{base}=\frac{14\sqrt3}{3}$

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.

$\text{side length}=2\left(\frac{14\sqrt3}{3}\right)$

Solve.

$\text{side length}=\frac{28\sqrt3}{3}$

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Question

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

Answer

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.

$\frac{18}{\sqrt3}=\frac{\text{base}}{1}$

Solve.

$\text{base}=\frac{18\sqrt3}{3}=6\sqrt3$

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.

$\text{side length}=2(6\sqrt3)$

Solve.

$\text{side length}=12\sqrt3$

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Question

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

Answer

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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Question

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

Answer

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.

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

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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Question

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

Answer

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.

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

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.

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

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Question

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

Answer

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.

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

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.

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

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Question

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

True or false: Quadrilateral is a rhombus.

Answer

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

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,

and

By transitivity,

.

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

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Question

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.

Answer

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

.

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.

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

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