Why do all exterior angles on a polygon add up to 360 when the interior changes depending on sides?

Question

Why do all exterior angles on a polygon add up to 360 when the interior changes depending on sides?

In: 6

Answer 1

Keep in mind the image of a [camera aperture closing](https://photographylife.com/wp-content/uploads/2018/01/Size-of-Aperture-Chart.jpg). Imagine extending each of the edges of a (convex) polygon into a ray like an aperture. The angles between the rays are the exterior angles. If we take this picture and “zoom out”, then this is like closing the aperture. But if we zoom out enough, the polygon disappear into basically just a dot with a bunch of lines going out in all directions like a totally closed aperture. Since these lines go all the way around this dot, the angles between them must add up to 360 degrees.

You can get the *interior* angles from this, but its just a bunch of algebra using this fact. For an intuitive reason for the 180(n-2) formula for interior angles we’ll do something else. It’s not exactly a proof, but it is a way to understand why the formula is as it is.

Say you have a polygon with n sides, and let’s say that you can stretch/compress sides and more them like they’re on hinges. This is the hand-wavy part, but it’s fine: Let’s say that as you change its shape by stretching sides or moving the hinges that the sum of the interior angles stays the same (as long as it remains convex). Lets then take any two vertices on the polygon and begin to pull, this will begin to flatten the polygon. Let’s pull really far, then what we’ll have is basically an extremely, extremely flat polygon with n sides (kinda like [this](https://ds055uzetaobb.cloudfront.net/brioche/uploads/cxSgLAgJWR-group-1.png) but with n sides instead of just 3). Because its so flat, we can just pretend that it *is* totally flat. Then the interior angle of the two vertices that we pulled will be zero because it is infinitely flat. But the interior angles of every other angle will have been stretched so that they are completely open, or 180 degrees. Since there are 2 with angle 0 and n-2 with angle 180 the total sum is 180(n-2) + 2*0 = 180(n-2).