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: Λv1 + Λv2 + Λv3 + Λv4

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: