The measure of an exterior angle of a regular decagon is degrees?

Answer 1

360

Answer 2

Short answer: The answer is 36. To get this, you simply take 360 degrees (the sum of all of the exterior angles) and divide it by how many angles there are (10).

There is some confusion as to the definition of an exterior angle. What it is: The exterior angle is defined as the angle between one side of a regular polygon and an imaginary line extended from a side adjacent to the first side. To find the measure of a regular polygon's external angle, you can either a) subtract the measure of one internal angle from 180; or b) divide 360 by the number of sides in the polygon (as explained above). This makes the internal and external angles supplementary or equal to 180 degrees when added together.

All exterior angles on a simple polygon, when added together, equal 360 degrees.

What it is not: Many people believe that the external angle is the entire outward reflex angle (greater than 180 degree) which can be found by subtracting the internal angle from 360 degrees. These two angles would be conjugate (equal to 360 degrees when added together). It has very little practical application outside of geometry theory itself. (I'm going to get hammered for that -- I said "little" not "absolutely none.")

How come I need to know this stupid stuff? First of all, you need it to pass your test. I assume you are reading this answer because you are cramming for a geometry test, right? Secondly, if you are going into anything that requires precision drawing (graphic design, drafting, construction, et al) then you need this because it is a cool shortcut to draw a regular polygon.

Finding the angle to draw without external angles: Count the number of sides. Let's say there are 10 sides (as in our decagon). Subtract 2. Multiply by 180. (Durnit -- the numbers are getting big now). Divide the big number by the original number of sides (10). That's the degrees of your interior angle. Here's your formula where x=[interior angle] and a=[number of sides]:

x = [180(a-2)] / a

x = [180(10-2) / 10

x = [180(8)] / 10

x = 1440 / 10

x = 144 degrees per angle.

Now draw your shape. Blech. Messy, big numbers.

Now try it with exterior angle shortcut: Count the number of sides. Divide 360 by the number of sides (the exterior angle). Subtract the answer from 180. There's the interior angle. Simple:

x = 180 - (360/a)

x = 180 - (360/10)

x = 180 - 36

x = 144 degrees per angle.

We got to the same answer with smaller numbers and fewer steps. You're welcome. :)

Answer 3

144