# Geometry/Chapter 7/Lesson 1

## Introduction

1. Names of polygons
2. Interior Angles (of a polygon)
3. Exterior Angles (of a polygon)

There will also be a

1. Polygon Chart for you to memorize.
2. A small question section.

## Names of Polygons

Names Sides
Triangle 3-sides
Pentagon 5-sides
Hexagon 6-sides
Heptagon 7-sides
Octagon 8-sides
Nonagon 9-sides
Decagon 10-sides
Dodecagon 12-sides
Other [number]-gon

## Interior Angles

Interior angles are angles within [inside] a polygon. In this section of the lesson, we will show you how to calculate individual interior angles and the sum of the interior angles. For example, in a "triangle", the total sum of the interior angles is 180. We got this because we figured out that each individual interior angle is 60.

You can figure out the angle degree of each interior angle by doing (note: n is the number of sides. In a triangle, there are 3 sides): ${\displaystyle {\tfrac {(n-2)(180)}{n}}}$. To find the total sum of interior angles, do: ${\displaystyle (n-2)(180)}$.

In a hexagon (6 sides) and a decagon (10 sides), if you do the equation for individual interior angles, ${\displaystyle {\tfrac {(n-2)(180)}{n}}}$, you get ${\displaystyle 120}$ and ${\displaystyle 144}$. And so on for every other polygon.

The total sums for individual interior angles are, by following ${\displaystyle (n-2)(180)}$, ${\displaystyle 720}$ and ${\displaystyle 1440}$.

### Interior Angle Sum Theorem

If a convex polygon has n (variable for: sides) and s represents the total sum of the interior angles, then:

${\displaystyle S=180(n-2)}$

The number of sides minus 2 gives you the number of triangles in the convex polygon, which you then times it by "180".

## Exterior Angles

An exterior angle, notebook definition, is "an angle formed by one side of the triangle and the extension of an adjacent side of the triangle"[1]. Pointers:

1. The total sum of exterior angles is ${\displaystyle 360}$.
2. To find the sum of individual exterior angles, simply do: ${\displaystyle {\tfrac {360}{n}}}$.

For example, in a octagon (8 sides), the total sum of exterior angles for this polygon is 360. Now, for the individual exterior angles: ${\displaystyle 360/8}$. This equals to ${\displaystyle 45}$.

### Exterior Angle Sums Theorem

If a polygon is convex, then the sum of the measures of the exterior angles, one at each vertex, is ${\displaystyle 360}$.

## Questions

### Question #1

1. Question: If an interior angle of a regular convex polygon is ${\displaystyle 140}$ degrees, how many sides does it contain and what is the name of this polygon?

Answer: An interior angle is supplementary with its exterior angle (linear pair), so we simply do: ${\displaystyle 180-140=40}$. After getting ${\displaystyle 40}$, we then do: ${\displaystyle 360/40}$, to give us the number of sides. Our answer is ${\displaystyle 9}$. This polygon is named: Nonagon.

### Question #2

1. Question: How many sides does a polygon have if the sum of the interior angles is ${\displaystyle 9540}$ degrees?

Answer: Crazy number, ain't it? But no worries: This is still an easy question to solve. Divide 9540 by 180 to give you the number of triangles in this polygon (We are using 180 because all of the interior angles of a triangle equal 180. We need the number of triangles in this polygon in order to times it by 180). 9540 divided by 180 gives "53". So, there are 53 triangles in this polygon. Now that you have 53, add 2. This will give you the number of sides: 55. So the answer is 55.

Double check:

• ${\displaystyle {(55-2)(180)}}$
• ${\displaystyle {(53)(180)}}$
• ${\displaystyle {9540}}$

This shows that in a polygon with 55 sides, the total interior angle sum is 9540. Therefore, a polygon with 55 sides has an interior angle sum of 9540.