Sum of all exterior angles of a polygon

To help you see what the sum of all exterior angles of a polygon is, we will use a square and then a regular pentagon. Since it is very easy to see what the sum is for a square, we will start with the square.

Notice that an exterior angle is formed by a side of the square and an extension of an adjacent side. For example in the figure below, angle x, angle y, angle z, and angle w are all exterior angles.

Each interior angle in a square is equal to 90 degrees. Notice that an interior angle plus the adjacent exterior angle is equal to 180 degrees.

Interior angle  +  adjacent exterior angle = 180 degrees.

Exterior angles

In fact, the sum of ( the interior angle plus the exterior angle ) of any polygon always add up to 180 degrees. This is so because when you extend any side of a polygon, what you are really doing is extending a straight line and a straight line is always equal to 180 degrees.

For example, 90 degrees + w = 180 degrees

90 degrees - 90 degrees + w = 180 degrees - 90 degrees

0 + w = 90 degrees

w = 90 degrees

Since there are 4 exterior angles, 4 x 90 degrees = 360 degrees.

Sum of all exterior angles of a polygon: pentagon

Sum of the exterior angles of a polygon

In the figure or pentagon above, we use a to represent the interior angle of the pentagon and we use x,y,z,v, and w to represents the 5 exterior angles.

To find the measure of the interior angle of a pentagon, we just need to use this formula.

[(n - 2 ) 180] / n 

Since n is equal to 5, [(n - 2 ) 180] / n = [(5 - 2) 180] / 5 = [3 x 180] / 5 = 540 / 5 = 108

Again, interior angle  +  adjacent exterior angle = 180 degrees.

108 degrees + adjacent exterior angle = 180 degrees

180 degrees - 180 degrees + adjacent exterior angle = 180 degrees

0 + adjacent exterior angle = 180 degrees

Adjacent exterior angle = 180 degrees

Since there are 5 exterior angles, 5 x 72 = 360 degrees.

It does not matter how many sides the polygon has, the sum of all exterior angles of a polygon is always equal to 360 degrees. 

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