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exterior angles of polygon gif

Polygons: Formula and Examples

Exterior Angles and Interior Angles

Interior Angle Sum Theorem

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

The sum of the measures of the interior angles of a convex polygon with n sides is $ (n-2)\cdot180^{\circ} $

Shape Formula Sum Interior Angles
$$ \red 3 $$ sided polygon
(triangle)
$$ (\red 3-2) \cdot180 $$ $$ 180^{\circ} $$
$$ \red 4 $$ sided polygon
(quadrilateral)
$$ (\red 4-2) \cdot 180 $$ $$ 360^{\circ} $$
$$ \red 6 $$ sided polygon
(hexagon)
$$ (\red 6-2) \cdot 180 $$ $$ 720^{\circ} $$
Problem 1

What is the total number degrees of all interior angles of a triangle?

180°

You can also use Interior Angle Theorem:$$ (\red 3 -2) \cdot 180^{\circ} = (1) \cdot 180^{\circ}= 180 ^{\circ} $$

Problem 2

What is the total number of degrees of all interior angles of the polygon ?

360° since this polygon is really just two triangles and each triangle has 180°

You can also use Interior Angle Theorem:$$ (\red 4 -2) \cdot 180^{\circ} = (2) \cdot 180^{\circ}= 360 ^{\circ} $$

Problem 3

What is the sum measure of the interior angles of the polygon (a pentagon) ?

Use Interior Angle Theorem:$$ (\red 5 -2) \cdot 180^{\circ} = (3) \cdot 180^{\circ}= 540 ^{\circ} $$

Problem 4

What is sum of the measures of the interior angles of the polygon (a hexagon) ?

Use Interior Angle Theorem: $$ (\red 6 -2) \cdot 180^{\circ} = (4) \cdot 180^{\circ}= 720 ^{\circ} $$

Video Tutorial

on Interior Angles of a Polygon

Definition of a Regular Polygon:

A regular polygon is simply a polygon whose sides all have the same length and angles all have the same measure. You have probably heard of the equilateral triangle, which are the two most well-known and most frequently studied types of regular polygons.

Examples of Regular Polygons

Regular Hexagon Regular Hexagon
Regular Pentagon Regular Hexagon
More on regular polygons here .

Measure of a Single Interior Angle

Shape Formula Sum interior Angles
Regular Pentagon $$ (\red 3-2) \cdot180 $$ $$ 180^{\circ} $$
$$ \red 4 $$ sided polygon
(quadrilateral)
$$ (\red 4-2) \cdot 180 $$ $$ 360^{\circ} $$
$$ \red 6 $$ sided polygon
(hexagon)
$$ (\red 6-2) \cdot 180 $$ $$ 720^{\circ} $$

What about when you just want 1 interior angle?

In order to find the measure of a single interior angle of a regular polygon  (a polygon with sides of equal length and angles of equal measure) with n sides, we calculate the sum interior anglesor $$ (\red n-2) \cdot 180 $$ and then divide that sum by the number of sides or $$ \red n$$.

The Formula
The measure of any interior angle of a regular polygon with $$ \red n $$ sides is

$ \text {any angle}^{\circ} = \frac{ (\red n -2) \cdot 180^{\circ} }{\red n} $

Example 1

Let's look at an example you're probably familiar with-- the good old triangle $$\triangle$$ . Now, remember this new rule above only applies to regular polygons. So, the only type of triangle we could be talking about is an equilateral one like the one pictured below. Equilateral triangle picture
You might already know that the sum of the interior angles of a triangle measures $$ 180^{\circ}$$ and that in the special case of an equilateral triangle, each angle measures exactly $$ 60^{\circ}$$.
Equilateral triangle picture angle labelled $ \text{Using our new formula} \\ \text {any angle}^{\circ} = \frac{ (\red n -2) \cdot 180^{\circ} }{\red n} \\ \text{ For a triangle , (} \red 3 \text{ sides)} \\ \frac{ (\red 3 -2) \cdot 180^{\circ} }{\red 3} \\ \frac{ (1) \cdot 180^{\circ} }{\red 3} \\ \frac{180^{\circ}} {\red 3} = \fbox{60} $

So, our new formula for finding the measure of an angle in a regular polygon is consistent with the rules for angles of triangles that we have known from past lessons.

Example 2

To find the measure of an interior angle of a regular octagon, which has 8 sides, apply the formula above as follows: $ \text{Using our new formula} \\ \text {any angle}^{\circ} = \frac{ (\red n -2) \cdot 180^{\circ} }{\red n} \\ \frac{(\red8-2) \cdot 180}{ \red 8} = 135^{\circ} $

Finding 1 interior angle of a regular Polygon

Problem 5

What is the measure of 1 interior angle of a regular octagon?

Substitute 8 (an octagon has 8 sides) into the formula to find a single interior angle

poly1
Problem 6

Calculate the measure of 1 interior angle of a regular dodecagon (12 sided polygon)?

Substitute 12 (a dodecagon has 12 sides) into the formula to find a single interior angle

poly2
Problem 7

Calculate the measure of 1 interior angle of a regular hexadecagon (16 sided polygon)?

Substitute 16 (a hexadecagon has 16 sides) into the formula to find a single interior angle

poly2
challenge problem Challenge Problem

What is the measure of 1 interior angle of a pentagon?

This question cannot be answered because the shape is not a regular polygon. You can only use the formula to find a single interior angle if the polygon is regular!

Consider, for instance, the irregular pentagon below.

You can tell, just by looking at the picture, that $$ \angle A    and    \angle B $$ are not congruent.

pentagon irregular

The moral of this story- While you can use our formula to find the sum of the interior angles of any polygon (regular or not), you can not use this page's formula for a single angle measure--except when the polygon is regular.

How about the measure of an exterior angle?

Exterior Angles of a Polygon

Formula for sum of exterior angles:
The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°.

Measure of a Single Exterior Angle

Formula to find 1 angle of a regular convex polygon of n sides = a


Exterior Angles of Triangle

$$ \angle1 + \angle2 + \angle3 = 360° $$


Exterior Angles of Polygon

$$ \angle1 + \angle2 + \angle3 + \angle4 = 360° $$


Exterior Angles of Pentagon

$$ \angle1 + \angle2 + \angle3 + \angle4 + \angle5 = 360° $$

Practice Problems

Problem 8

Calculate the measure of 1 exterior angle of a regular pentagon?

Substitute 5 (a pentagon has 5sides) into the formula to find a single exterior angle

poly2
Problem 9

What is the measure of 1 exterior angle of a regular decagon (10 sided polygon)?

Substitute 10 (a decagon has 10 sides) into the formula to find a single exterior angle

poly2
Problem 10

What is the measure of 1 exterior angle of a regular dodecagon (12 sided polygon)?

Substitute 12 (a dodecagon has 12 sides) into the formula to find a single exterior angle

poly2
challenge problem Challenge Problem

What is the measure of 1 exterior angle of a pentagon?

This question cannot be answered because the shape is not a regular polygon. Although you know that sum of the exterior angles is 360, you can only use formula to find a single exterior angle if the polygon is regular!

Consider, for instance, the pentagon pictured below. Even though we know that all the exterior angles add up to 360 °, we can see, by just looking, that each $$ \angle A    and    \angle B $$ are not congruent..

Determine Number of Sides from Angles

It's possible to figure out how many sides a polygon has based on how many degrees are in its exterior or interior angles.

Problem 11

If each exterior angle measures 10°, how many sides does this polygon have?

Use formula to find a single exterior angle in reverse and solve for 'n'.

equation
Problem 12

If each exterior angle measures 20°, how many sides does this polygon have?

Use formula to find a single exterior angle in reverse and solve for 'n'.

equation
Problem 13

If each exterior angle measures 15°, how many sides does this polygon have?

Use formula to find a single exterior angle in reverse and solve for 'n'.

equation
challenge problem Challenge Problem

If each exterior angle measures 80°, how many sides does this polygon have?

There is no solution to this question.

When you use formula to find a single exterior angle to solve for the number of sides , you get a decimal (4.5), which is impossible. Think about it: How could a polygon have 4.5 sides? A quadrilateral has 4 sides. A pentagon has 5 sides.

challenge