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## Geometry (all content)

### Course: Geometry (all content) > Unit 3

Lesson 5: Angles with polygons# Sum of the exterior angles of a polygon

Learn a simple and elegant way to find the sum of the exterior angles of any convex polygon. You will see how to redraw the angles adjacent to each other and form a circle. Then you will discover that the sum of the exterior angles is always 360 degrees. Watch this video to master this important skill in geometry. Created by Sal Khan.

## Want to join the conversation?

- Is 360 degrees for all polygons ?(145 votes)
- You've been lied to.

It will actually work for any polygon, as long as you remember to use negative numbers for the concave angles. The answer is always 360°, and you can prove it by drawing a shape something like https://goo.gl/photos/zFSQs2XwDxwqKGwZ8 (sorry for the terrible picture). The -90° makes up for the two extra 45°s, and so it comes out even.(11 votes)

- I was confused by the definition of "exterior angles".

If the interior angle of one corner is, say, 90 degrees (like a corner in a square) then shouldn't the exterior angle be the whole outside of the angle, such as 270? Why is only 90 degrees counted for the exterior angle of a corner instead of 270?

In other words, exterior corners look like they are always greater than 180, but we subtract the 180. Why?(76 votes)- It's just the way exterior angles are defined.

From the wikipedia article: "an exterior angle (or external angle) is an angle formed by one side of a simple, closed polygon and a line extended from an adjacent side."

See: http://en.wikipedia.org/wiki/Exterior_angle(11 votes)

- is a star considered as a convex polygon?(5 votes)
- No, it is concave because it has an angle greater than 180 degrees( also known as a reflex angle.(10 votes)

- What is the definition of a convex polygon?(4 votes)
- A
**convex polygon**is a many-sided shape where*all*interior angles are less than 180' (they point outward).

Examples of convex polygons:

- all triangles

- all squares

An octagon with equal sides & angles (like a stop sign) is a convex polygon; the pentagons & hexagons on a soccer ball are convex polygons too.

There are also**concave polygons**, which have*at least one*internal angle that is greater than 180' (points inward).

Examples of concave polygons:

- a star

- a cross

- an arrow

To tell whether a shape is a**convex polygon**, there's an easy shortcut: just look at the pointy parts (or "vertices"). If*every single one*of the points sticks out, then the polygon is convex!

Hope this helps!(9 votes)

- At the very start of the video, Sal references to a video done "several videos ago". Could someone please link the video he's talking about?(3 votes)
- I'm pretty sure this is the video he is talking about: https://www.khanacademy.org/math/geometry/hs-geo-foundations/hs-geo-polygons/v/sum-of-interior-angles-of-a-polygon(3 votes)

- The sum of interior angles of a regular polygon is 540°. Calculate the size of each exterior angle. *(3 votes)
- You need to know four things. the sum of all exterior angles equal 360, allexterior angles are the same, just like interior angles, and one exterior angle plus one interior angle combine to 180 degrees. Finally, the sum of interior angles is found with the formula 180(n-2) where n is the number of angles. since it tells us the sum we can find the number of angles.

180(n-2)=540

n-2 = 3

n = 5

So five corners, which means a pentagon. this means there are 5 exterior angles. since they all have to add to 360 you can divide 360/5 = 72. You can also check by adding one interior angle plus 72 and checking if you get 180.

total interior angle is 540, there are 5 angles so one angle is 108. 108+72 = 180 so this confirms that one exterior angle is 72 degrees.

Let me know if aything didn't make sense.(2 votes)

- What is the meaning of anticlockwise?(2 votes)
- it is the same as counter-clockwise, which is the opposite of the direction the hands of a clock go.

Or if you start at the top of a circle, and go down and around to the left.(3 votes)

- What is concave and convex? (If you see this and you know the answer please answer. Thank you!)(1 vote)
- A concave lens "caves in". We can extend this to geometry as well. Concave polygons will have a part or parts that are sticking inwards, instead of being outwards. A Concave polygon could be a boomerang shape, while a convex polygon would be any regular polygon, since it doesn't cave in. The formal definition for a polygon to be concave is that at least one diagonal (distance between vertices) must intersect with a point that isn't contained in the polygon.(5 votes)

- So, how would you find the exterior angle of a reflex angle? And does the sum of the exterior angles of a concave polygon always equal 360, or does it equal 540?

Can someone please help me!!(2 votes)- I found this at https://en.wikipedia.org/wiki/Angle#Types_of_angles

"An exterior angle measures the amount of rotation one has to make at a vertex to trace out the polygon. If the corresponding interior angle is a reflex angle, the exterior angle should be considered negative.", so if the reflex angle is 220, it appears that the exterior angle would be -40 degrees, this would mean that the 360 would still be the sum of exterior angles. I learned something new.(2 votes)

- A convex polygon is a polygon that is not caved in. Have you ever seen an arrow that looks like this: ➢? That is a concave polygon. This:∇ is a convex polygon. If you still don't "get it" I would look at this link for more information (and pictures) because this is kind of hard to explain. http://en.wikipedia.org/wiki/Convex_and_concave_polygons

Hope this helps!(4 votes)

## Video transcript

Several videos ago, I had a figure that looked something like this. I believe it was a pentagon or a hexagon. And what we had to do is figure out the sum of the particular exterior angles of the hexagon. So it would've been this angle, we should call A, this angle B, C, D, and E. And the way that we did it the last time, we said, "Well, A is going to be 180 degrees "minus the interior angle that is supplementary to A." And then we did that for each of the angles. And then we figured out we were able to algebraically manipulate it. We were able to figure out what the sum of the interior angles were using dividing it up into triangles, and then use that to figure out the exterior angles. And it was a bit of an involved process. What I want to show you in this video is there's actually a pretty simple and elegant way to figure out the sum of these particular external angles, exterior angles I should say, of this polygon. And it actually works for any convex polygons when you're picking these particular external, these particular exterior angles, I should say. And so the way to think about it is you can just redraw the angles. So let's just draw each of them. So let me draw this angle right over here. We could call it angle A or maybe the measure of this angle is A, either way. Let me draw it right over here. So it's going to be, this is going to be a congruent angle, right over here. It's going to have a measure of A. Now let me draw angle B, angle B. And I'm going to draw adjacent to angle A. And what you could do is think about it. Maybe if we drew a line right over here, if we drew a line right over here that was parallel to this line, then the measure of this angle right over here would also be B, because this obviously is a straight line. It would be like a transversal. These are corresponding angles. So if we wanted to draw the adjacent angle be adjacent to A, you could do it like that or the whatever angle this is, its measure is B. Then now it's adjacent to A, and now let's draw the same thing for C. We could draw a parallel line to that right over here. And then this angle would also be C. And if we want it to be adjacent to that, we could draw it right over here. So that angle is C. So C would look something like this. C would look something like that. Then we can move on to D. Once again, let me do that in a different color. You could do D. D could be right over here, or you could shift it down over here to look like that. Or you could shift it over here to look like that. If we just kept thinking about parallel... If all of these lines were parallel to each other, so let's just draw D like this. Let's just draw D like this. D like this. So this line once again's gonna be parallel to that line. And then finally, you have E. Finally, you have angle E. And once again, you could draw a line. You could draw a line that is parallel to this right over here. Right over here, and this right over here would be angle E, or you can draw it right over here. You would draw it right over here. And when you see it drawn this way, it's clear that when you add up the measure, this angle A, B, C, D, and E, you're going all the way around the circle. Either way, you could be going... You could be going clockwise, or you could be going counter-clockwise, but you're going all the way around the circle. And so the sum of these angles are just going to be... So A plus B, plus C, plus D, plus E is just going to be 360 degrees. And this will actually work as I said, for any convex polygon. When I say convex polygon, I mean one that's not dented inwards, one that's kind of... So just to be clear, what I'm talking about... It would work for any polygon that is kind of... I don't want to say regular. Regular means it has the same sides and same angles, but it's not dented. So this is a convex polygon. This right here is a concave polygon. So let me draw it this way. So this right over here would be a concave, would be a concave polygon. I'm gonna draw it as a having the same number of sides. So I just kind of dented these two sides right over there. And did I do that right? Let me see. Let me do it the same number of sides. So I want to do that, that, that, that, and then I know that's the same side over there. Let me draw it like that. And then like that. This has one, two, three, four, five, six sides. This has one, two, three, four, five, six sides. This is concave. Sorry, this is convex. This is a convex polygon. This is a concave polygon. And the way I remember it is kind of caved inwards. And so what we just did would apply to any. If we're trying to find these particular external, exterior angles of any convex polygon, I afraid, I apologize ahead of time if I've confused them all, because I have a feeling that I might've. This applies to any convex polygon. And so once again, if you take this angle and add it to this angle, and add it to this angle, add it to this angle, add it to that angle, and add it to that angle. And I'm not implying that they're all going to be the same. I just drew it that way. I could show you that they are different angles. So I could say that one in green and that one in some other color, I think you get the idea. They can all be different, but when you if you shift the angles like this you'll see that they just go around the circle. So once again, they'll just add up to 360 degrees.