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Sum of the exterior angles of convex polygon

More elegant way to find the sum of the exterior angles of a convex polygon
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Sum of the exterior angles of convex polygon

Discussion and questions for this video
Is 360 degrees for all polygons ?
The sum of the exterior angles of any convex polygon is 360°
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?
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You can think of exterior angles as how many degrees you need to turn at each corner if you're tracing the line around the polygon.

When you walk around the block, you turn 90° at each corner, not 270°. The same applies for any shape. If you were to walk around a hexagon, you would turn only 60° at each corner.
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How do you find the exterior angles of a conCAVE polgon, then?
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Actually, it is 360°, according to http://en.wikipedia.org/wiki/Exterior_angle
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Can you have a negative angle?
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Yes, if you say that angles measured counterclockwise are positive and those measured clockwise are negative : )
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is a star considered as a convex polygon?
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A star is not a convex polygon. Draw a line between two star points, the line is outside the star.
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In the example in the video, would the answer be 360 degrees? If so would it always be 360 degrees? If not what would the answer be and how can you find it?
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Yes, the answer is 360 degrees and will always be 360 degrees for any convex polygon. Good job on figuring that out!
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In a polygon with n sides at what point does n+1 equal a circle?
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A circle is not a polygon. Polygons are composed of straight sides. A circle does not have straight sides, it has curving sides.
what is a convex polygon?
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A convex polygon is a polygon in which no line containing a side of the polygon intersects the interior of the polygon.
It's opposite would be a concave polygon, where a line containing one side of the polygon DOES intersect the interior of the polygon.
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The more sides a convex polygon, (with sides equal in length), the more it will look like a circle. Therefore, is a circle a polygon with 360 sides which would be equal to sides/points?

Stated another way -
Since there is no theoretical limit to a polygon at what point does the circum-circle of a polygon equal a circle? If so, is a circle a polygon?
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A polygon must have straight sides. It cannot have curving sides. However, as you have shown, a curved side can be made from an infinite amount of straight sides. Combining this together there are only two logical conclusions that I can think of. One, a polygon cannot have an infinite amount of sides. It can have 10 sides. It can have 10 million sides. It can have a million million sides. But it cannot have infinity sides. The other logical conclusion is that infinity sides does not make a curve ( I have no argument to back this up ).
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Why didn't Sal define convex?
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3:29 Oh wait, never mind...

From now on, I think I'll wait until the end of the video before commenting.
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How do you find out the measure of the interior angles with only information about the measure of exterior angles?
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Exterior angles and interior angles are supplementary. Thus
Interior Angle = π - exterior angle (in radians)
Interior Angle = 180° - exterior angle (in degrees)
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I get that the sum of the exterior angles of a convex polygon add up to 360 degrees but I have a feeling the way Sal explained it in the video is more on the intuitive side. It is not a demonstration of sorts. Is this concept maybe one of those "intuitively obvious" things that cannot really be proved but is taken as given nonetheless?
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Well the way Sal explained it does not provide a proof for all convex polygons, or 'the general case'. He just made this to show us the intuitive way of finding the sum of the exterior angles.

However I'm pretty sure there is a way to prove this, though I haven't seen the proof myself. Good question though.
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then how do you find the interior angles of a convex polygon?
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Do any of your teachers make you do two column proofs like a chart that seperates and says statement and reason because mine does and its really annoying
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I agree that the proofs can be annoying, but I think they are about the most valuable skills that you can learn in geometry. No, you will most likely not need to be able to prove which triangles in a complex drawing are congruent. However, the skill of being able to think your way through a problem, systematically, step-by-step, logically, without making unwarranted assumptions, is a very valuable skill.

So, I'd agree it is annoying at first, but it is a skill worth knowing.
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So for every polygon, even if it had 800 sides, would the exterior angles always add up to 360, assuming it is a convex polygon?
Also, what about a concave polygon? Does the part that is caved inwards have an external angle, although it is over 180 degrees? Or is it just the other exterior angles that are not caved in that add up to 360 degrees?
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Are concave polygons also add up to 360 degrees?
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no, the external angles of a concave polygon do not add up to 180... for example, take a 6 sided concave polygon with one internal reflex angle.. the sum of the external angles in this case turns out to be 540=180*5+360*1-(4*180) where the (4*180) is sum of internal angles!
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How do I find the specific angle of an exterior angle?
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em um polígono regular basta dividir 360° pelo numero de lados (360°/n)...
em um polígono irregular prolongue o angulo interno e subtraia-lo de 180° (angulo interno -180°)
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In astronomy, what causes -- in the case of regular convex polygons ("regular" convex structures) -- the sum of the interior angles to be infinite and the sum of the exterior ones to be constant?!
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a Platonic solid is a regular, convex polyhedron with congruent faces of regular polygons and the same number of faces meeting at each vertex. Five solids meet those criteria, and each is named after its number of faces.
What previous video is he referencing in the beginning?
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Sum of Interior Angles of a Polygon ?
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explain how you can determine if a polygon is concave
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If you can draw diagonals outside a shape, then that shape is a concave polygon.
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Are all exterior polygons 360 degrees?
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Yes, the sum of the exterior angles for ANY polygon is 360, making the very interesting formula:
S(Exterior Angles) = 360 degrees
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What is the difference between a convex and concave polygon?
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a regular polygon has 20 degrees, find the number of sides it has
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I am assuming that the sum of interior angles is 20 degrees.
lets take the number of sides as n
(n-2) * 180 = 20 degrees (sum of interior angles of a polygon having n sides is (n-2) * 180)
180*n - 360 = 20
n = 19/2
which is not possible so that means it is not a polygon instead for having a polygon the sum of interior angles must be equal or more than 180 degrees.
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At 4:35, Sal explains a way that he remembers concave and convex... Is there any other way?
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In Spanish _con_ means _with_ so concave can mean _with cave_. Another trick to determining concavity is to imagine an elastic band being stretched around the figure, does it touch all sides or does it have to span a gap. If it has to span a gap then the figure must be concave (having a cavity or cave).
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what is a convex polygon
?
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Here's a non-rigorous definition: If you can *always* draw a line from any point in the polygon to any other point in the polygon without going outside the polygon, it is convex.

More at Wikipedia: http://en.wikipedia.org/wiki/Convex_polygon
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does this apply to all polygons
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If they are convex, yes
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What about the formula: (n-2) x 180?
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It is used to find sum of the interior angles. N = number of sides.
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When I click on these links, it sends me to YouTube. Then, it says I haven't seen the video and I get no credit for it. It used to work before, but not anymore so how do I fix it?
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Does this apply to concave polygons ?
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No it does not. The rule that all the exterior angles of a polygon add up to 360 only works on convex polygons.
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is the polygon , means more than one sides , am a little confused??
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def Polygon: A closed plane figure bounded by three or more line segments. It comes from the Greek polygonos which means 'many-angled'. So any 2-dimensional object with three or more sides is a polygon (triangles, rectangles, octagons etc).
Does this apply for concave polygons as well?
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Yes, however, it sometimes requires using negative angles (the exterior angle of a reflex interior angle is negative).
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in the video the captions are in Chinese ....what do I do?
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Go on the caption setting and go on english
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So are all exteriors just 360 and then you divide the shape for interiors with triangles?
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The exterior angles will add to 360 on all CONVEX polygons. If the polygon is CONCAVE, the exterior angles might not add to 360. Sal shows the difference between a convex and concave polygon at 3:50

For all polygons, you can find the sum of the interior angles by multiplying 180 times two less than the number of sides. For example, a square (having four sides) would have an interior angle sum of 180 x (4 - 2) = 360.
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What exactly is a CONVEX polygon?
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Its the opposite of a CONCAVE polygon
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if the measure of an exterior angle of a regular polygon is 45 degrees then the polygon is A. a decagon
B.a an octagon
C.a pentagon
D.a square
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Why is there no sound in this video!
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your computer is probably messed up
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What is the major difference between Convex and Concave Polygon?
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A convex polygon can never have an interior angle that is greater than 180 degrees. A concave polygon can. That's the major difference. I hope this helps!
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I am slow at math, so bear with me if I get confused but I understand this video in bits and pieces. How do they decide where to draw the lines out to determine the exterior angle of the polygon? Why did he make corresponding angles with parallel lines (transversals) if we're not deducing each angle measurement and only finding out that because of some drawing it will always equal 360 for any convex polygon? Seems a little too indepth if the explanation is really that simple. If it is always 360 degrees, why do we need to know the drawing and adjacent angle explanations? That just threw me off. Can anyone explain that part as well? Thanks in advance.
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Why doesnt it work for a convave polygon?
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wouldn't an un closed polygon have an exterior angle something different than 360
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a "unclosed polygon" doesn't exsist
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The sum of the exterior angles of a polygon, including a regular, is 360 degrees, right? Or is it just for convex polygons?
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Just for convex.
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How do you know when to use this strategy and when to use the 180(s-2) strategy from the sum of interior angles of a polygon video?
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I think you can use that strategy all the time with the interior angles of any kind of polygon
example: triangle is 180(3sides -2)
square is 180(4-2)
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at 4:46 you just said it's true about convex polygans,what about cancave polygans?
I just think that it's true about concave polygans too,isn't it?
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Where is the video that sal referenced at 0:00
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When you know the number of triangles formed,how do you find the sum of the interior angles of a polygon
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You multiply the number of triangles by 180 and that's the sum of the interior angles.
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At 3:29, doesnt convex have a second meaning?
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What is a convex region
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how do i algabraically make a hexagon add up to 360?
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what is the exact equation for exterior angles
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add all the measures of the exterior angles and set them equal to 360
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How would you work out the exterior or interior angle of a concave polygon?
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This might be a stupid question but how can it equal 360 degrees? a perfect circle is 360 degrees so that would make every polygon a broken circle, but you cant square a circle? So if you have say a square that exterior angles = 360 degrees wouldnt it be able to have the same area as a circle?
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Anything that goes around that doesn't have a beginning or end = 360 degrees.
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does a concave polygon also have a measure of 60 degrees?
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Could you demonstrate that a regular convex polygon with infinite sides has a total sum of exterior angles that is infinite?

Watch the video ; it will help a lot.

Thank you :)
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how do you determine how to solve a regular polygon equation
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what do you mean by solving to find a regular polygon equation?
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Man, I'm learning this stuff in Honors Geometry right now.
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How do you know when you're supposed to add the interior, or the exterior angles. That's the only reason I can't pass this part. Please help!
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At 1:39,shouldn`t angle A be + to point B?
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Is the sum of exterior angles 360 for all polygons? And how about triangles?
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Triangles also have exterior angles of 360 for the sums. An equalateral triangle has 60˚ for each _interior_ angle. That means that each exterior angle _must_ be 120˚. 120*3 is 360. Hope this helped you understand better!
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im trying to find the values of the variables for the polygon, triangle.
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The polygon exterior angles theorem says that the sum of the measures of the exterior angles of a convex polygon one angle at each vertex is 360. What does one angle at each vertex mean?
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vertex are points, make those points into angles
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how to you find the exterior angles of a concave polygon ?
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Actually, it's the same as a convex polygon. The sum is also 360°
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wait, so is every polygon's exterior angle is 360 degrees?
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it's applicable only for CONVEX...don't forget that part!
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Is this true for all polygons that the exterior angles are equal to 360?
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Yes. All exterior angles add up to 360.
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Why don't this method work for Concave Polygon and why it only works for Convex Polygon? What is the reason behind it?
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so a concave polygon has one angle greater than 180 degrees?
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If any of the angles on the interior of the polygon are acute, it is concave
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Is there a better way to find out if a polygon is a convex polygon? If we don't know the angles in degrees, we can't be sure that it really does equal 360 degrees, can we? I think I'm confused, haha.
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In the video, Sal also did not know about the measurement of any triangle, but he was able to prove it anyways.
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Is geometry part of algebra?
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Some parts of geometry are a part of algebra, but fir the most part, geometry is geometry. You do use a lot of algebra in geometry in solving segments, tangents and equal angles. So, in a way, algebra s the language of geometry.
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I still don't get what convex and concave polygons are....
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Convex-this is less than 180 degrees.
Concave-think you are going inside a cave, more than 180 degrees.
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If a polygon has 7 edges does that mean that it's 900 degrees?
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You figure out the degrees a polygon has by using the formula (n-2)180.
When n = the number of edges. Heptagons (7-sided polygons), therefore would give the answer of:
(7-2)180
(5)180
900. So yes, its total number of degrees is 900.

You can then take the equation farther, and divide 900 by the number of sides (in this case, 7) to obtain the degrees of each angle in the heptagon.
900/7 = 128.571... (The formula to find this is (n-2)180 / n; n = # of sides.)
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Is 360 degrees for all polygons ?
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Yes every exterior angle on a convex polygon addes up to be 360.
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On any polygon you can probable extend the lines at any point where two sides meet to have the external a angles equal 360 degrees. Therefore, if you have six points of two lines meeting as shown in the current polygon used, wouldn’t the sum of all the external angles of that polygon be equal to 6 times 360 degrees?
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On any polygon you can probable extend the lines at any point where two sides meet to have the external a angles equal 360 degrees. Therefore, if you have six points of two lines meeting as shown in the current polygon used, wouldn’t the sum of all the external angles of that polygon be equal to 6 times 360 degrees?
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so a polygons exterior angles always add up to 360?
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Yes while the sum of the interior angles is found by taking the # of sides, subtracting 2, and then multiplying by 180 degrees (n-2)(180°). It has to be convex in orther for the exterior angles to add up to 360 degrees though.
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What would you do for a concave polygon
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Can someone please tell me like what the maxium sum of a polygon is? Thanks :)
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there is no maximum sum
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How do you find out whether the exterior angles are 360 degrees or something else?
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In any convex polygon the exterior angles should add up to 360 degrees.
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is it always 360?
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yes
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I still don't understand about concave and convex polygons. Can someone please take the time to explain to me?

-mdanivas, a.k.a. Pants
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What?

-mdanivas, a.k.a. Pants
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Report a mistake in the video
Example:

At 2:33, Sal said "single bonds" but meant "covalent bonds."

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