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


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

In: 6

Its a result of the fact that all interior angles in a polygon equals 180(n-2) where n is the number of angles.

Since each exterior angle is the inverse of its interior counterpart the num of those 2 angles is equal to 180.

I think this question is more about the deeper meaning of those angles.

If the exterior angles were just the angles on the outside of corners then they would also get bigger than 360. But instead they’re something a little more special and are measured from the line of action of the corner. This makes them measure the turn accumulated in the shape and if it’s a closed shape it will add to 360.

Imagine starting at a point in the middle of one of the polygon’s edges and walking one full loop around the polygon’s perimeter. When you return to your starting point, you will have turned a total of 360 degrees and be facing the same direction as when you started.

The exterior angles of a polygon measure the amount of turning that happens at each vertex. For a convex polygon, all turns are in the same direction (either all clockwise or all counterclockwise), so adding up the angles of the individual turns gives you the angle of the total turn you make over the entire loop, which we’ve established is 360 degrees.

For a non-convex polygon, some of the turns are in opposite directions so the exterior angles do not necessarily sum to 360. However, the rule works again if you treat angles in the clockwise direction as negative and angles in the counterclockwise direction as positive, or vice versa. If you track negative angles like this over a path that crosses over itself, you can actually get sums that are larger multiples of 360, like 720 or 1080.

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).

Idk if you have ever used the program MS logo, but imagine you are on a giant piece of paper and that you are drawing with your feet (like a turtle in said program).
When you make a regular polygon you are going to move some distance forward and some angle to the left (or right, but let’s keep it counter clockwise for simplicity). Then you move forward and repeat this process till you are back at the starting position (If you are back to where you started, you created a regular polygon, else it’s just a random shape).

Let’s recreate this motion that you used to create said polygon. The only difference this time is that you do not move forward but only turn. After the first step you have turned some angle, and so on and on. On the last step, you turn one final angle and are facing the original position, i.e. you turned 360°. This means that the sun of all left turns is 360.

By virtue of turning left to create the polygon, the angle you turned is the exterior angle of the polygon. Hence through this thought experiment (I think it’s called the spider problem) we have demonstrated that no matter the polygon, the sum of exterior angle is ALWAYS a full rotation (360°).

By basic geometry the interior angle is (180° – exterior angle). And the number of internal angles created is dependent on the number of polygonal sides you created ie the number of steps you took in this experiment. Hence it depends on the polygon.

Please lmk if you need elaboration

The total angles at every vertex of the polygon add up to 180. They have to, since the interior and exterior angles form a straight line by definition. So the sum of both interior and exterior angles of a polygon with *n* vertices is necessarily 180 * n

The interior angles add up to 180(n-2), or if you prefer, 180n – 360 (just distribute the 180 into the parentheses). So the remaining 360 have to go to the exterior angles.

Well, that’s not quite true, unless you make 2 important caveats:

– For concave angle, the exterior angle is negative, where you count the angle “backward”.

– The polygon cannot self-intersect.

Imagine you’re walking along the polygon. At each corner, you turn. The amount of turning you made is the exterior angle (negative turning means you turn away from the interior of the polygon). If you walk a full length of the polygon, turning at each corner, then you would have turned a full rotation, which means the total sum is 1 rotation, or 360 degree.

Why you turn a total of 1 round? Well, the number of round you turn is always an integer, because you face the same direction as when you started. And if you turn 0 round total, or at least 2, the polygon must self-intersect, which we don’t allow.

What about interior angle? Sum of interior and exterior is 180 degrees, always. So the total sum of them depends on the number of angles, hence number of sides. But exterior always add up to 360 degree, so the rest is interior.