# Angles in Shapes

## Angles of Polygons

To find the sum of angles in a polygon, you can simply use the following formula:

$Sum\quad of\quad angles\quad =\quad (n-2)\quad \times \quad 180^\circ$

where n is the number of sides

For example, a triangle has three sides therefore the total sum of angles is:

$Sum\quad of\quad angles\quad of\quad triangles = (3-2)\times\ 180^\circ = 180^\circ$

$Quadrilateral = (4-2)\times\ 180^\circ = 360^\circ$

This could go on and on...

$Pentagon = (5-2)\times\ 180^\circ = 540^\circ$

$Hexagon = (6-2)\times\ 180^\circ = 720^\circ$

In fact, a table could be made here...

 Polygon Sides Sum of Angles Triangle 3 180° Quadrilateral 4 360° Pentagon 5 540° Hexagon 6 720° Heptagon 7 900° Octagon 8 1080° Nonagon 9 1260° Decagon 10 1440°

...of course, the number of polygons does not stop here. However, there is an ever increasing pattern of 180° per one side added.

In addition, for any regular polygon, dividing the sum of angles by the number of sides (or vertices) would result in the angle of each vertex. A regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be convex or concave.

### Examples:

1. Find the sum of angles of a regular polygon that has 15 sides (pentadecagon):

Use the sum of angles formula:

$(n-2)\times 180^\circ \\ =(15-2)\times 180^\circ \\ = 2340^\circ$

2. Find the angle on a vertex of a regular pentadecagon:

Since the sum of angles is 2340°, then

$\cfrac { 2340^{ \circ } }{ 15 } =156^{ \circ }$

## Triangles

There are different types of triangles:

(courtesy of learnhive.net)

### Exterior Angle of a Triangle

The exterior angle in any triangle is equal to the sum of the two opposite interior angles. That is,

$a^\circ+c^\circ=d^\circ$

Think: Why is the exterior angles equal to the sum of the two opposite interior angles?