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

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.




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  }





There are different types of triangles:

(courtesy of

Exterior Angle of a Triangle


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


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



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