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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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8 Diagnostic Tests 250 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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

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.

Possible Answers:

$12\:cm$

$20.8\:cm$

Cannot be determined

$12\sqrt3\:cm$

$6\:cm$

Correct answer:

$12\:cm$

Explanation:

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.

Find_the_length

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, n = 6 .

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

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

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, $\frac{6\sqrt3\:cm}{x}$ .

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

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

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

$x=6\:cm$

Now $6\:cm$ 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 $12\:cm$ .

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

Suppose the perimeter of a hexagon is 24a+15b . What is the side length of the hexagon?

Possible Answers:

4a+4b

$4a+\frac{3}{2}b$

$4a+\frac{5}{2}b$

$4a+\frac{5}{3}b$

18a+9b

Correct answer:

$4a+\frac{5}{2}b$

Explanation:

Write the perimeter formula for hexagons.

P=6s

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

24a+15b=6s

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

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Example Question #3 : How To Find The Length Of The Side Of A Hexagon

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

Possible Answers:

$\frac{5}{3}$

$\frac{5}{2}$

$\frac{3}{2}$

Correct answer:

$\frac{5}{2}$

Explanation:

Write the formula for finding the perimeter of a hexagon.

P=6s

Substitute the perimeter and solve for the side length.

15=6s

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

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Example Question #4 : How To Find The Length Of The Side Of A Hexagon

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

Possible Answers:

$1+\sqrt3$

$2+\sqrt3$

$\frac{1+\sqrt3}{2}$

$18+36\sqrt3$

$\frac{1+2\sqrt3}{2}$

Correct answer:

$\frac{1+2\sqrt3}{2}$

Explanation:

Write the perimeter formula for a hexagon.

P=6s

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

If the perimeter of a regular hexagon is 15a+16b , what must be the side length of the hexagon?

Possible Answers:

$\frac{20}{3}ab$

$\frac{40}{9}ab$

$\frac{31}{6}ab$

$\frac{5}{3}a+\frac{8}{3}b$

$\frac{5}{2}a+\frac{8}{3}b$

Correct answer:

$\frac{5}{2}a+\frac{8}{3}b$

Explanation:

Write the formula for the perimeter of a hexagon.

P=6s

Substitute the perimeter and solve for the side.

15a+16b=6s

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

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Example Question #1 : 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:

$12\sqrt3$

$8\sqrt3$

$7\sqrt3$

$10\sqrt3$

Correct answer:

$8\sqrt3$

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.

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

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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Example Question #1 : 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:

$10\sqrt3$

$11\sqrt3$

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

$\frac{28\sqrt3}{3}$

Correct answer:

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

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.

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

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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Example Question #1 : 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:

$\frac{22\sqrt3}{3}$

$\frac{44\sqrt3}{3}$

$\frac{88\sqrt3}{3}$

$\frac{11\sqrt3}{3}$

Correct answer:

$\frac{44\sqrt3}{3}$

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.

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

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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Example Question #2 : 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:

$\frac{28\sqrt3}{3}$

$\frac{56\sqrt3}{3}$

$\frac{29\sqrt3}{3}$

$7\sqrt3$

Correct answer:

$\frac{28\sqrt3}{3}$

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.

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

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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Example Question #1 : 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:

$24\sqrt3$

$6\sqrt3$

$12\sqrt3$

$18\sqrt3$

Correct answer:

$12\sqrt3$

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.

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

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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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

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

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

Example Question #3 : How To Find The Length Of The Side Of A Hexagon

Example Question #4 : How To Find The Length Of The Side Of A Hexagon

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

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

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

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

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

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

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