Intermediate Geometry : How to find an angle in a hexagon

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

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

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

Possible Answers:

\(\displaystyle 180^{\circ}\)

\(\displaystyle 60^{\circ}\)

\(\displaystyle 720^{\circ}\)

\(\displaystyle 120^{\circ}\)

cannot be determined

Correct answer:

\(\displaystyle 120^{\circ}\)

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:

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

In this case, \(\displaystyle n = 6\).

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

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

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

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

\(\displaystyle 120^{\circ}\)

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:

\(\displaystyle 120\)

\(\displaystyle 156\)

\(\displaystyle 180\)

\(\displaystyle 720\)

\(\displaystyle 88\)

Correct answer:

\(\displaystyle 120\)

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.

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

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

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

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

\(\displaystyle \frac{720}{6}=120\)

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

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

True or false: Quadrilateral \(\displaystyle CAGO\) is a rectangle.

Possible Answers:

True

False

Correct answer:

False

Explanation:

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

Hexagon 2

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

\(\displaystyle m \angle CAG = 60 ^{\circ }\).

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

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

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

Possible Answers:

False

True

Correct answer:

True

Explanation:

A regular polygon with \(\displaystyle N\) sides has \(\displaystyle N\) congruent angles, each of which measures 

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

Setting \(\displaystyle N = 6\), the common angle measure can be calculated to be

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

The statement is therefore true.

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

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

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 \(\displaystyle 360^{\circ }\). Each exterior angle of a regular hexagon has the same measure, so if we let \(\displaystyle t\) be that common measure, then

\(\displaystyle 6t = 360 ^{\circ }\)

Solve for \(\displaystyle t\):

\(\displaystyle 6t \div 6 = 360 ^{\circ } \div 6\)

\(\displaystyle t = 60^{\circ }\)

The statement is false.

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

Given: Hexagon \(\displaystyle HEXAGO\).

\(\displaystyle m \angle H = 108 ^{\circ }\)

True, false, or undetermined: Hexagon \(\displaystyle HEXAGO\) is regular.  

Possible Answers:

Undetermined

True

False

Correct answer:

False

Explanation:

Suppose Hexagon \(\displaystyle HEXAGO\) is regular. Each angle of a regular polygon of \(\displaystyle N\) sides has measure

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

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

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

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

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:

\(\displaystyle 18^{\circ }\)

\(\displaystyle 20^{\circ }\)

\(\displaystyle 16^{\circ }\)

\(\displaystyle 15^{\circ }\)

\(\displaystyle 12^{\circ }\)

Correct answer:

\(\displaystyle 18^{\circ }\)

Explanation:

The sum of the exterior angles of any polygon, one at each vertex, is \(\displaystyle 360^{\circ }\). 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.

\(\displaystyle 360\div 20 = 18\) 

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:

\(\displaystyle 95^{\circ }, 105^{\circ }, 115^{\circ }, 125^{\circ }, 135^{\circ }, 145^{\circ }\)

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

\(\displaystyle 110^{\circ },110^{\circ }, 110^{\circ }, 110^{\circ }, 110^{\circ }, 170^{\circ }\)

\(\displaystyle 118^{\circ },119^{\circ }, 120^{\circ }, 120^{\circ }, 121^{\circ }, 122^{\circ }\)

\(\displaystyle 120^{\circ }, 120^{\circ }, 120^{\circ }, 120^{\circ }, 120^{\circ }, 120^{\circ }\)

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 \(\displaystyle 180 (6-2)=720^{\circ }\)

Add the angle measures in each group.

\(\displaystyle 120+120+120+120+120+120=720\)

\(\displaystyle 95+105+115+125+135+145=720\)

\(\displaystyle 110+110+110+110+110+170=720\)

\(\displaystyle 118+119+120+120+121+122=720\)

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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