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Intermediate Geometry : How to find an angle in a hexagon

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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:

$180^{\circ}$

$60^{\circ}$

$720^{\circ}$

$120^{\circ}$

cannot be determined

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

What is the interior angle of a regular hexagon if the area is 15?

Possible Answers:

120

156

180

720

Correct answer:

120

Explanation:

The area has no relevance to find the angle of a regular hexagon.

There are 6 sides in a regular hexagon. Use the following formula to determine the interior angle.

$\theta=(n-2)\cdot 180$

Substitute n=6 sides to determine the sum of all interior angles of the hexagon in degrees.

$\sum\theta=(6-2)\cdot 180=720$

Since there are 6 sides, divide this number by 6 to determine the value of each interior angle.

$\frac{720}{6}=120$

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Example Question #1 : How To Find An Angle In 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 rectangle.

Possible Answers:

True

False

Correct answer:

False

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

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

The angles of a rectangle must measure $90^{\circ }$ , so it has been disproved that Quadrilateral CAGO is a rectangle.

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

True or false: Each of the six angles of a regular hexagon measures $120^{\circ }$ .

Possible Answers:

False

True

Correct answer:

True

Explanation:

A regular polygon with sides has congruent angles, each of which measures

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

Setting N = 6 , the common angle measure can be calculated to be

$\frac{(6-2)180^{\circ }}{6} = \frac{4 \cdot 180^{\circ }}{6} = \frac{720^{\circ }}{6} = 120^{\circ }$

The statement is therefore true.

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Example Question #2 : How To Find An Angle In A Hexagon

True or false: Each of the exterior angles of a regular hexagon measures $80^{\circ }$ .

Possible Answers:

False

True

Correct answer:

False

Explanation:

If one exterior angle is taken at each vertex of any polygon, and their measures are added, the sum is $360^{\circ }$ . Each exterior angle of a regular hexagon has the same measure, so if we let be that common measure, then

$6t = 360 ^{\circ }$

Solve for :

$6t \div 6 = 360 ^{\circ } \div 6$

$t = 60^{\circ }$

The statement is false.

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

Given: Hexagon HEXAGO .

$m \angle H = 108 ^{\circ }$

True, false, or undetermined: Hexagon HEXAGO is regular.

Possible Answers:

Undetermined

True

False

Correct answer:

False

Explanation:

Suppose Hexagon HEXAGO is regular. Each angle of a regular polygon of sides has measure

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

A hexagon has 6 sides, so set N = 6 ; each angle of the regular hexagon has measure

$\frac{(6-2)180^{\circ }}{6} = \frac{ 4 \cdot 180^{\circ }}{6} = 120^{\circ }$

Since one angle is given to be of measure $108 ^{\circ }$ , the hexagon cannot be regular.

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Example Question #3 : How To Find An Angle In A Hexagon

What is the measure of one exterior angle of a regular twenty-sided polygon?

Possible Answers:

$18^{\circ }$

$20^{\circ }$

$16^{\circ }$

$15^{\circ }$

$12^{\circ }$

Correct answer:

$18^{\circ }$

Explanation:

The sum of the exterior angles of any polygon, one at each vertex, is $360^{\circ }$ . In a regular polygon, the exterior angles all have the same measure, so divide 360 by the number of angles, which, here, is 20, the same as the number of sides.

$360\div 20 = 18$

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Example Question #2 : How To Find An Angle In A Hexagon

Which of the following cannot be the six interior angle measures of a hexagon?

Possible Answers:

$95^{\circ }, 105^{\circ }, 115^{\circ }, 125^{\circ }, 135^{\circ }, 145^{\circ }$

All of these can be the six interior angle measures of a hexagon.

$110^{\circ },110^{\circ }, 110^{\circ }, 110^{\circ }, 110^{\circ }, 170^{\circ }$

$118^{\circ },119^{\circ }, 120^{\circ }, 120^{\circ }, 121^{\circ }, 122^{\circ }$

$120^{\circ }, 120^{\circ }, 120^{\circ }, 120^{\circ }, 120^{\circ }, 120^{\circ }$

Correct answer:

All of these can be the six interior angle measures of a hexagon.

Explanation:

The sum of the interior angle measures of a hexagon is $180 (6-2)=720^{\circ }$

Add the angle measures in each group.

120+120+120+120+120+120=720

95+105+115+125+135+145=720

110+110+110+110+110+170=720

118+119+120+120+121+122=720

In each case, the angle measures add up to 720, so the answer is that all of these can be the six interior angle measures of a hexagon.

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Intermediate Geometry : How to find an angle in a hexagon

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

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

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

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

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

Example Question #2 : How To Find An Angle In A Hexagon

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

Example Question #3 : How To Find An Angle In A Hexagon

Example Question #2 : How To Find An Angle In A Hexagon

All Intermediate Geometry Resources

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