2.5 A Polygon’s Angle Sum
An angle sum is the sum of all the angles in a geometric figure. 

Example:
The angle sum in this quadrangle.
Angle sum: Λv_{1} + Λv_{2} + Λv_{3} + Λv_{4} 


Angle Sum for a Triangle
Draw and try to measure and add all the angles in a triangle. What do you get? How big is the angle sum? Is the angle sum always the same?
We can try a little experiment to find out how big the angle sum of a triangle is. 


1. Draw and mark circular arc and angle names. Cut out the triangle. 
2. Cut out the corners, angles. 



3. Draw a line and mark a point. Then lay the angles’ vertexes against the point which we have done in this picture. 
4. Together they create a straight angle which is always 180º. So we have shown that the angle sum in a triangle is 180º.




Example:
Calculate the angle v. Here we know that one angle and the symbol for a right angle (90°). We know that the angle sum of a triangle is 180º.
60º + 90º + v = 180º
Then v = 180º  90º  60º.
v = 30º 



Angle sum of a Quadrangle
An entire rotation in a circle is 360°. If you divide the circle into 4 equal pieces then each fourth of a rotation 360° ÷ 4 = 90° which is the measurement of a right angle. 

Here is a quadrangle with four right angles (rectangle). A rectangle always has a angle sum of 90° + 90° + 90° + 90° = 360°. 

If you divide a quadrangle into two equal parts with a diagonal, you get two triangles.
Every such triangle has an angle sum of 180°. Because there are two triangles, the angle sum in a quadrangle is: 180° + 180° = 360°.



Naturally, you don’t need the quadrangle to be a rectangle for you to be able to divide it into two triangles.





Angle Sum in a Pentagon.
A pentagon can be divided into three triangles by using two lines. Each triangle has an angle sum of 180°.
The angle sum for a pentagon is therefore:
3 · 180° = 540° 




As you can see, the angle sum increase by 180° for every corner/triangle.
The formula for calculating the angle sum in a polygon where the number of corners is equal to n is: 

