﻿ Polygons: Formula for Exterior Angles and Interior Angles, illustrated examples with practice problems on how to calculate..

# Polygons: Formula and Examples

Exterior Angles and Interior Angles

### Interior Angle Sum Theorem

#### What is true about the sum of angles inside a polygon (ie interior angles)?

The sum of the measures of the interior angles of a convex polygon with n sides is $(n-2)180^{\circ}$

Examples:
• Triangle or ( '3-gon')
• Quadrilateral which has four sides ( ' 4-gon')
• sum of interior angles: $$(\red 4-2) 180 = 360^{\circ}$$
• Hexagon which has six sides ( '6-gon')
• sum of interior angles: $$(\red 6-2) 180 = 720^{\circ}$$

### 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. The most well known example of a regular polygon is the equilateral triangle

### Measure of a Single Interior Angle

#### 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 just divide the sum of the interior angles or (n-2) × 180 by the number of sides or n

The Formula

An interior angle of a regular polygon with n sides is $\frac{ (\red n -2) \cdot 180^{\circ} }{\red n}$

Example:

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

180°

360° since the polygon on the left is really just two triangles and each triangle has 180°

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

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

### Finding 1 interior angle of a regular Polygon

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

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

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

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.

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 =

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

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

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

### Practice Problems

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

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

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

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.

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

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

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

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.

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