Angles in Shapes

angleshapes

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

How about a 4-sided polygon (also called quadrilaterals)?

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):

 

Show Answer

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:

 

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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?

Quadrilaterals

 

A quadrilateral is any four-sided figure, and the sum of angles in any interior angles in a quadrilateral is 360°.

The family of special quadrilaterals are shown below:

(courtesy of skwirk.com)