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Math resources Geometry Angles in polygons

Sum of ext. angles of a polygon

Sum of exterior angles of a polygon

Here you will learn about how to find the sum of exterior angles of a polygon using a single exterior angle and use this knowledge to solve problems.

Students will first learn about this topic as a part of high school geometry.

What is the sum of exterior angles of a polygon?

The sum of the exterior angles of a polygon is always $360^{\circ}$ regardless of the number of sides the polygon has. This is also referred to as the polygon exterior angle sum theorem.

The exterior angles are angles between a polygon and the extended line from the vertex of the polygon.

Sum of exterior angles of a polygon $=360^{\circ}$

Sum of exterior angles of a polygon image 1 US

The interior angle and exterior angle at a vertex form a straight line so they add to $180^{\circ}.$

Sum of exterior angles of a polygon image 2 US

For a regular polygon, each exterior angle is the same and is calculated using the formula

E=\cfrac{360}{n}

where $E$ represents one exterior angle and n is the number of sides of the regular polygon.

As always, the angle $\theta$ is measured in degrees, $^{\circ}.$

What is the sum of exterior angles of a polygon?

Common Core State Standards

How does this relate to high school math?

High school: Geometry (HS.G.CO.C.10)
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to $180^{\circ};$ 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.

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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!

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How to solve problems involving the sum of exterior angles of a polygon

In order to solve problems involving exterior angles of a polygon:

Identify the number of sides of any polygon(s) and note whether they are regular or irregular polygons.
Identify what the question is asking and recall the sum of exterior angles.
Use the known information and any correct formula to solve.

Sum of exterior angles of a polygon examples

Example 1: finding the size of a single exterior angle for a regular polygon

Find the size of a single exterior angle for a regular hexagon.

Sum of exterior angles of a polygon image 3 US

Identify the number of sides of any polygon(s) and note whether they are regular or irregular polygons.

A hexagon has $6$ sides. These sides are all regular, and therefore all exterior angles are equal.

2Identify what the question is asking and recall the sum of exterior angles.

The question is asking to find the size of one exterior angle. The sum of exterior angles for a polygon is $360^{\circ}.$

3Use the known information and any correct formula to solve.

360 \div 6=60

The size of each exterior angle is $60^{\circ}.$

Example 2: finding an exterior angle given an interior angle for an irregular polygon

An irregular octagon has one interior angle of size $130^{\circ}.$ What is the size of the adjacent exterior angle?

Identify the number of sides of any polygon(s) and note whether they are regular or irregular polygons.

An irregular octagon will have $8$ sides.

Identify what the question is asking and recall the sum of exterior angles.

As adjacent means next to, you are being asked to find the size of the exterior angle where the interior angle on the line is known.

Use the known information and any correct formula to solve.

Angles on a straight line add to $180^{\circ}$ and the interior angle is $130^{\circ}.$

$180-130=50^{\circ}$

The exterior angle will be $50^{\circ}.$

Example 3: interior + exterior angle = 180°

Find the measure of angle $x.$

Sum of exterior angles of a polygon image 4 US

Identify the number of sides of any polygon(s) and note whether they are regular or irregular polygons.

An irregular hexagon has $6$ sides.

Identify what the question is asking and recall the sum of exterior angles.

You need to find the interior angle $x.$ It is an irregular polygon so the exterior angles and interior angles are not all equal.

The sum of exterior angles for a polygon is $360^{\circ}.$

Use the known information and any correct formula to solve.

Find the missing exterior angle of the polygon first.

As one interior angle is $90^{\circ},$ the exterior angle at this vertex is

$180-90=90^{\circ}.$

Subtract the $5$ known exterior angles from $360^{\circ}$ to determine the unknown exterior angle.

$360-(30+60+20+50+90)=110^{\circ}$

The interior angle $+$ the exterior angle must equal $180^{\circ}$ .

So $x=180-110=70^{\circ}.$

Example 4: finding the number of sides given the exterior angle of a regular polygon

An exterior angle of a regular polygon is $20^{\circ}.$ How many sides does the polygon have?

Identify the number of sides of any polygon(s) and note whether they are regular or irregular polygons.

The polygon has an unknown number of sides, or $n$ sides.

It is a regular polygon, therefore all exterior angles are equal.

The sum of exterior angles for a polygon is $360^{\circ}.$

Identify what the question is asking and recall the sum of exterior angles.

You need to find the number of sides, $n.$

You know the sum of the exterior angles is $360^{\circ}$ and that each exterior angle is equal because it is a regular polygon.

Use the known information and any correct formula to solve.

\begin{aligned}20\times{n}&=360 \\\\ n&=18 \end{aligned}

The polygon has $18$ sides.

Example 5 : finding the number of sides given the interior angle of a regular polygon

The size of each interior angle of a regular polygon is $150^{\circ}.$ How many sides does the polygon have?

Identify the number of sides of any polygon(s) and note whether they are regular or irregular polygons.

The polygon has an unknown number of sides, or $n$ sides.

It is a regular polygon, therefore all exterior angles are equal.

Identify what the question is asking and recall the sum of exterior angles.

You need to find the number of sides.

The sum of the exterior angles is $360^{\circ}$ and each exterior angle is equal because it is a regular polygon.

The sum of an interior and an exterior angle is $180^{\circ}.$

Use the known information and any correct formula to solve.

If the interior angle is $150^{\circ}$ then the exterior angle will be $180-150=30^{\circ}.$

The number of sides can therefore be calculated by $360\div{30}=12.$

The polygon has $12$ sides.

Example 6: multi step problem involving interior and exterior angles

The size of each interior angle of a regular polygon is $11$ times the size of each exterior angle. Work out the number of sides the polygon has.

Identify the number of sides of any polygon(s) and note whether they are regular or irregular polygons.

The polygon has an unknown number of sides, or $n$ sides.

It is a regular polygon, therefore all exterior angles are equal.

Identify what the question is asking and recall the sum of exterior angles.

You are given the number of sides of the polygon.

Other Information to consider:

Total of exterior angles $=360^{\circ}$
Interior $+$ Exterior angle $=180^{\circ}$
$11\times\text{Interior angle}=\text{Exterior angle}$

Use the known information and any correct formula to solve.

You can call each of the interior angles $x.$

Since $11\times\text{interior angle}=\text{exterior angle},$ each exterior angle can be called, $11x.$

Therefore,

\begin{aligned}x+11x&=180 \\\\ 12x&=180 \\\\ x&=15 \end{aligned}

The size of one exterior angle is $15^{\circ}.$

The number of sides of the polygon is $360 \div 15=24.$

The polygon has $24$ sides.

Teaching tips for sum of exterior angles of a polygon

Provide students with a step-by-step proof that shows that the sum of exterior angles of any polygon will always add up to $360^{\circ}.$

Instead of providing students with worksheets to practice with, allow students to work in pairs or small groups to practice. This allows students the ability to collaborate and explain their thinking and reasoning in their own words.

Get students to investigate the interior and exterior angles of polygons by drawing and measuring these angles.

Easy mistakes to make

Confusing exterior and interior angles
The exterior angle of a triangle is the angle between the side and the extension of an adjacent side. Here the interior angle (internal angle) is $60^{\circ},$ so the exterior angle (external angle) must be $120^{\circ}.$

Incorrectly assuming all the angles are the same size
The sum of exterior angles will always be $360$ degrees, regardless of the number of sides. More sides does not mean the sum of all exterior angles will be greater than a polygon with less shapes.

Misunderstanding regular vs. irregular polygons
Students may believe that irregular polygons follow another set of rules than regular polygons. Regardless if the polygon is regular or irregular, the sum of the measures of the exterior angles will always be $360$ degrees.

Practice questions for sum of exterior angles of a polygon

90^{\circ}

60^{\circ}

180^{\circ}

270^{\circ}

Exterior angles of a polygon add up to $360^{\circ}.$

A regular quadrilateral has $4$ interior angles equal in size, so the four exterior angles are equal.

This means you can divide $360$ by $4$ to get the solution.

360\div{4}=90^{\circ}

45^{\circ}

60^{\circ}

40^{\circ}

135^{\circ}

Exterior angles of a polygon add up to $360^{\circ}.$

A regular octagon has $8$ interior angles equal in size, so the eight exterior angles are equal.

This means you can divide $360$ by $8$ to get the solution.

360\div{8}=45^{\circ}

90^{\circ}

40^{\circ}

140^{\circ}

280^{\circ}

Exterior angles of a polygon add up to $360^{\circ}.$

A regular nonagon has $9$ interior angles equal in size, so the nine exterior angles are equal.

This means you can divide $360$ by $9$ to get the solution.

360\div{9}=40^{\circ}

$12$ sides

$20$ sides

$30$ sides

$32$ sides

Exterior angles of a polygon add up to $360^{\circ}.$ This means you can divide $360$ by $12$ to get the solution.

$12$ sides

$20$ sides

$16$ sides

$18$ sides

Exterior angles of a polygon add up to $360^{\circ}.$ This means you can divide $360$ by $20$ to get the solution.

360\div{20}=18

The polygon has $18$ sides.

125^{\circ}

40^{\circ}

55^{\circ}

140^{\circ}

The four known exterior angles will be $55^{\circ},$ since angles on a straight line add to $180.$

This means the fifth exterior angle will be $140^{\circ}$ because exterior angles add up to $360^{\circ}.$

360-(55\times{4})=140^{\circ}

Using angles on a straight line once more leads us to find out the missing angle is $40^{\circ}.$

49^{\circ}

131^{\circ}

139^{\circ}

41^{\circ}

The four known exterior angles are $40^{\circ}, \, 64^{\circ}, \, 28^{\circ},$ and $89^{\circ}.$

As one interior angle is $90^{\circ}$ and the sum of the interior and exterior angle at a vertex is $180^{\circ},$ the exterior angle at this vertex is $180-90=90^{\circ}.$

As exterior angles add up to $360°,$

360-(40+64+28+89+90)=49^{\circ}

Using angles on a straight line once more leads us to find out the missing angle is $180-49=131^{\circ}.$

Sum of exterior angles of a polygon FAQs

What is a polygon?

A polygon is a two dimensional shape with at least three sides, where the sides are all straight lines. There are two key types of polygons.

$1)$ A regular polygon is where all angles are equal size and all sides of a polygon are equal length (a square), and $2)$ An irregular polygon where all angles are not equal size and/or all sides are not equal length (a trapezoid).

What are the exterior angles of a polygon?

The sum of exterior angles of any polygon is always $360^{\circ},$ regardless of the number of sides the polygon has.

For a regular polygon, the exterior angles can be found using the formula

$\text{exterior angle}=\cfrac{360^{\circ}}{n},$ where $n,$ is the number of sides of the polygon.

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

The sum of interior angles of a polygon can be found by using the interior angles formula: $\text{sum of interior angles}=(n-2)\times{180}^{\circ}.$

Unlike the sum of exterior angles, the sum of interior angles does change based on the number of sides of a polygon.

The next lessons are

Congruence and similarity
Transformations
Mathematical proof
Trigonometry

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