Angles of a Hexagon - Math Steps, Examples & Questions

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Angles of a hexagon

Here you will learn about angles of a hexagon, including finding the sum of the interior angles and solving problems involving interior angles and exterior angles.

Students will first learn about angles of a hexagon as part of geometry in high school.

What are angles of a hexagon?

Angles of a hexagon are the angles in a six-sided polygon (2D shape).

There are different types of hexagons:

A regular hexagon has six equal side lengths, six vertices, and six equal interior angles.

An irregular hexagon has $6$ sides which are not all equal, and $6$ interior angles which are not all equal.

The sum of all interior angles of a hexagon is $720^{\circ}.$

The sum of all exterior angles of a hexagon is $360^{\circ}.$

[FREE] Angles Check for Understanding Quiz (Grade 4)

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[FREE] Angles Check for Understanding Quiz (Grade 4)

Use this quiz to check your grade 4 students’ understanding of angles. 10+ questions with answers covering a range of 4th grade angles topics to identify areas of strength and support!

DOWNLOAD FREE

Interior and exterior angles of a hexagon

We need to be able to solve problems involving angles of a hexagon.

To do this, you need to work with the interior and exterior angles of a hexagon.

A pair of interior and exterior angles of any polygon add to $180^{\circ}$ because they form a straight line. They are supplementary angles.

Interior angles of a hexagon are the angles inside the 2D shape, formed when two sides of the shape meet.

We can find the sum of the interior angles of a hexagon, labeled $I,$ by using the formula:

$I=(n-2)\times{180},$ where $n$ is the number of sides.

A hexagon has $6$ sides, so $n=6.$

$\begin{aligned}I&=(6-2)\times{180} \\\\ &=4\times{180} \\\\ &=720^{\circ} \end{aligned}$

The formula above uses the relationship between the angles in a triangle and the number of triangles a polygon can be split into.

To find an interior angle of a regular hexagon, you can divide the sum of the interior angles by the number of sides, in this case, $6.$

Step-by-step guide: Interior angles of a polygon

Exterior angles of a hexagon are the angles between the hexagon and the extended line from the next side.
The sum of the exterior angles of any polygon is always $360^{\circ}.$ To find each of the exterior angles of a regular hexagon, labeled $E,$ you can use the formula

$E=\cfrac{360}{n},$

where $n$ is the number of sides.

For a regular hexagon, $n=6$ so $E=360\div{6}=60^{\circ}$

Alternatively, if you know the size of the exterior angle of a polygon, you can use a rearrangement of the above formula to identify the number of sides the polygon has

n=\cfrac{360}{E}

Step-by-step guide: Sum of exterior angles of a polygon

What are angles of a hexagon?

Common Core State Standards

How does this relate to high school math?

High school – Geometry – Congruence (H.G.CO.C.10)
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to $180^{\circ}\text{;}$ base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

How to solve problems involving angles of a hexagon

In order to solve problems involving angles of a hexagon:

Identify if the hexagon is regular or irregular.
Identify what the question is asking for.
Solve the problem using the information you have gathered.

Angles of a hexagon examples

Example 1: finding the exterior angle of a regular hexagon

Below is a regular hexagon. Find the angle marked $x.$

Identify if the hexagon is regular or irregular.

The question tells you that this is a regular hexagon. This means the sides of the hexagon are equal and the interior angles are equal.

2Identify what the question is asking for.

The question wants you to find the angle $x,$ which is one of the six exterior angles of this hexagon.

3Solve the problem using the information you have gathered.

To find the size of the exterior angles of a regular hexagon you can use the formula

$E=\cfrac{360}{n},$ where $n$ is the number of sides.

So, the missing angle can be calculated as

x=\cfrac{360}{6}=60^{\circ}.

So, the exterior angle of a regular hexagon is $60^{\circ}.$

Example 2: finding the sum of the interior angles of a hexagon

What is the sum of the interior angles of a hexagon?

Identify if the hexagon is regular or irregular.

The sum of the interior angles of a hexagon will be the same whether it is a regular or irregular hexagon.

Identify what the question is asking for.

The question asks you to calculate the sum of the internal angles of a hexagon.

Solve the problem using the information you have gathered.

To calculate the sum of the interior angles of a hexagon,

$I=(n-2)\times{180},$ where $n$ is the number of sides.

So, the interior angle of a hexagon is

\begin{aligned}I&=(6-2)\times{180} \\\\ &=4\times{180} \\\\ &=720^{\circ} \end{aligned}

Example 3: finding an interior angle of a regular hexagon

Below is a regular hexagon shape. Find the angle marked $x.$

Identify if the hexagon is regular or irregular.

The question tells us that this is a regular hexagon.

Identify what the question is asking for.

The question wants us to find the angle $x,$ which is one of the six interior angles of this hexagon.

Solve the problem using the information you have gathered.

To find the size of an interior angle of any hexagon, you need to calculate the sum of the interior angles first.

To calculate the sum of the interior angles of any polygon, you can use the formula

$I=(n-2)\times{180},$ where $n$ is the number of sides.

So, the interior angle of a hexagon can be calculated as

\begin{aligned}I&=(6-2)\times{180} \\\\ &=4\times{180} \\\\ &=720^{\circ} \end{aligned}

Now that you know the angles inside a hexagon add up to $720^{\circ},$ you can divide this by the number of sides, in this case $6,$ to find the size of one interior angle.

So, the missing angle can be calculated as

$x=\cfrac{720}{6}=120^{\circ}.$

Example 4: finding an interior angle of an irregular hexagon

Below is a polygon. Find the size of the missing angle $x.$

Identify if the hexagon is regular or irregular.

The interior angles of the hexagon are not equal, so it must be irregular.

Identify what the question is asking for.

The question asks you to find the missing interior angle.

Solve the problem using the information you have gathered.

The sum of interior angles for a hexagon is $720^{\circ}.$

Next, subtract the interior angles you know to find the missing value.

$x=720-(94+131+103+122+108)=162^{\circ}.$

Example 5: finding an exterior angle of an irregular hexagon

Below is an irregular polygon. Find the size of the missing angle $x.$

Identify if the hexagon is regular or irregular.

The question states that the polygon is irregular.

Identify what the question is asking for.

The question asks you to find the missing exterior angle.

Solve the problem using the information you have gathered.

To find the size of the exterior angle, $x,$ you need to find the missing interior angle that corresponds to it, labeled $y$ on the diagram below.

The sum of interior angles for a hexagon is $720^{\circ}.$

Next, subtract the interior angles you know to find the missing value.

$x=720-(101+142+118+142+113)=104^{\circ}$

Now that you know the value of the interior angle corresponding to $x,$ you can use the fact that the interior and exterior angles of all polygons add to $180^{\circ}$ because they form a straight line, to calculate the exterior angle.

So, $x=180-104=76^{\circ}.$

Example 6: solving angle problems involving hexagons

The diagram shows a tessellation from a section of floor tiles.

The shape is formed by two regular polygons.

Find the size of the angle labeled $x.$

Identify if the hexagon is regular or irregular.

The question states that both polygons in the diagram are regular polygons.

Identify what the question is asking for.

The questions asks us to find the size of the angle $x,$ which is formed on the exterior of the two regular hexagons.

Solve the problem using the information you have gathered.

To find the size of the angle, $x,$ you need to find the missing interior angles that meet it at the point.

The sum of interior angles for a hexagon is $720^{\circ}.$

Next, divide this by the number of sides, in this case $6,$ to find the size of one of the interior angles.

$\cfrac{720}{6}=120^{\circ}$

Now that you know that the two interior angles are $120^{\circ}$ each, you can use the angle fact ‘angles around a point add up to $360^{\circ}$ ’ to find the angle $x.$

So, $x=360-(120+120)=120^{\circ}.$

Teaching tips for angles of a hexagon

Show examples of hexagons in real life, such as honeycombs, tiles, and nuts/bolts.

Ensure that students have a strong understanding of the properties of hexagons. Mention that they can be regular (equal angles and equal lengths) or irregular. Discuss specific types such as concave hexagons and convex hexagons. Review concepts such as the area of a regular hexagon and perimeter of a regular hexagon.

Provide quizzes and worksheets to test understanding, with a mix of multiple-choice, short answer, and drawing problems. Utilize math tutors as needed.

Have students use calculators to verify calculations with both regular and irregular hexagons.

Easy mistakes to make

Not identifying regular and irregular polygons
It is important to correctly identify whether a polygon is regular or irregular, using the information given in the question or on the diagram. Do not assume one or the other.

Using the formulae incorrectly
Whether a polygon is regular or irregular will determine how the formulae for working with interior and exterior angles can be used.

For example, the formula $E=\cfrac{360}{n},$ where $n$ is the number of sides.

It can only be used to find the size of exterior angles of regular polygons, where all exterior angles are equal.

Practice angles of a hexagon questions

60^{\circ}

120^{\circ}

720^{\circ}

180^{\circ}

We can find the size of an exterior angle of any regular polygon using the formula

$E=\cfrac{360}{n},$ where $n$ is the number of sides.

So, the size of exterior angles of a regular hexagon can be calculated as $\cfrac{360}{6}=60^{\circ}.$

60^{\circ}

120^{\circ}

720^{\circ}

1080^{\circ}

We can find the sum of the interior angles of any polygon using the formula

$I=(n-2)\times{180},$ where $n$ is the number of sides.

As $n=6$

\begin{aligned}I&=(6-2)\times{180} \\\\ &=4\times{180} \\\\ &=720^{\circ} \end{aligned}

120^{\circ}

60^{\circ}

643^{\circ}

77^{\circ}

To find the value of one interior angle of an irregular polygon, you first need to calculate the sum of the interior angles of the polygon.

We can find the sum of the interior angles of any polygon using the formula

$I=(n-2)\times{180},$ where $n$ is the number of sides.

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

Now that you know the angles inside the hexagon add up to $720^{\circ},$ you can subtract the interior angles you know to find the missing value.

x=720-(142+130+71+152+148)=77^{\circ}

720^{\circ}

120^{\circ}

60^{\circ}

118^{\circ}

The interior and exterior angles of any polygon at a vertex add up to $180^{\circ},$ so, $x=180-62=118^{\circ}.$

628^{\circ}

60^{\circ}

92^{\circ}

720^{\circ}

The sum of interior angles for a hexagon is $720^{\circ}.$

Subtract the interior angles you know to find the missing value.

y=720-(157+112+128+99+132)=92^{\circ}

60^{\circ}

120^{\circ}

30^{\circ}

180^{\circ}

Each exterior angle of a regular hexagon is $\cfrac{360}{6}=60^{\circ}.$

Angles of a hexagon FAQs

What is the sum of the interior angles of a hexagon?

The sum of the interior angles of a hexagon is $720^{\circ}.$

What is the measure of each interior angle in a regular hexagon?

In a regular hexagon, where all interior angles are equal, each interior angle measures $120^{\circ}.$

What is the sum of the exterior angles of a hexagon?

The sum of the exterior angles of any polygon, including a hexagon, is always $360^{\circ}.$

What is the measure of each exterior angle in a regular hexagon?

In a regular hexagon, where all exterior angles are equal, each exterior angle measures $60^{\circ}.$

How do lines of symmetry affect the angles of a hexagon?

In a regular hexagon, the six lines of symmetry ensure all interior $(120^{\circ})$ and exterior $(60^{\circ})$ angles are equal by dividing the hexagon into identical parts.

In irregular hexagons, lines of symmetry, if present, divide the shape and angles equally along the symmetry line.

The next lessons are

Congruence and similarity
Transformations
Mathematical proof

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