Sum of the exterior angles of convex polygon

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

Sum of the exterior angles of convex polygon

Discussion and questions for this video
Nope! As Sal told at 3:25 it will work for any Convex Polygon, it is clear theat it is only liable for Convex Polygons. Notice, he didn't said that is for all polygons. So don' t get confused with Convex Polygon and all polygon.........

Hope this will help you out of this!
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?
As I think about this now, it makes sense:

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.
How do you find the exterior angles of a conCAVE polgon, then?
In a polygon with n sides at what point does n+1 equal a circle?
A circle is not a polygon. Polygons are composed of straight sides. A circle does not have straight sides, it has curving sides.
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?
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 ).
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?
No, it is concave because it has an angle greater than 180 degrees( also known as a reflex angle.
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.
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.
At 3:10 Sal said something like clockwise and counterclockwise..
What does he mean by clockwise and counterclockwise ?
Clockwise on the clock rotation goes as top-right-down-left.
Counterclockwise on the clock rotation goes as top-left-down-right.

Most clocks go Clockwise.
It's the way the hands on an analog clock move. if you have a line segment from the center of a circle going straight up, when it moves to the right going on all the way around however far that is clockwise, if it goes left that is counterclockwise.
What is the difference between convex and concave shapes ?
"Convex and Concave Polygons. Every polygon is either convex or concave. The difference between convex and concave polygons lies in the measures of their angles. For a polygon to be convex, all of its interior angles must be less than 180 degrees."
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?
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.
Can a polygon have a higher angle than 360?
Yes you can, but I don't think you would need an angle more than 360 degrees.
How do you find out the measure of the interior angles with only information about the measure of exterior angles?
Exterior angles and interior angles are supplementary. Thus
Interior Angle = π - exterior angle (in radians)
Interior Angle = 180° - exterior angle (in degrees)
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
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.
To find out only one exterior angle(of a convex polygon), do you do 360/ number of sides?
For anyone who is confused. Basically what Sal was trying to say was that every Convex polygon has all of its angles adding up to 360 degrees.
Sal is *not* saying "that every Convex polygon has all of it angles adding up to 360 degrees." He is saying that the sum of all the *exterior* angles of a convex polygon is 360 degrees. The sum of the interior angles depends on the number of the sides of the polygon. The concept of the sum of the interior angles of a polygon is covered in other videos.
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?
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.
Hope this helps!
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.
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!
If you can draw diagonals outside a shape, then that shape is a concave polygon.
I am just curious, but what happens when you have a concave polygon?
The sum is still 360 but you have to subtract the negative angles caused by the concave parts. It is a bit trickier.
The sum of the exterior angles of a polygon with vertices at A,B,C,D,E,F,G,H,I would be 360 degrees. This is a property of the sum of the exterior angles of a polygon.
is there a difference between the sum of the interior angles of a square and a trapezoid and for all quadrilaterals
No any quad that isn't concave (dented) would make 2 triangles so it would be 180 * 2 or 360 interior.
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?!
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.
Why is is that the sum of the exterior angles for all polygons is 360 degrees no matter haw many sides the polygon has?
Because all a polygon is is a circle with sides. We define a circle as not having any sides, and having its points all equally distanced away from one point. We've given circles 360 degrees to better explain them, and as a result, any closed figures' exterior angles must add to 360.
Would students benefit from learning that the sum of exterior angles is tau radians?
When adding up the exterior angles of convex polygons, it doesn't matter how many angles there are. The sum of them is ALWAYS 360 degrees. I'm sorry, I cannot watch the video, because it isn't working for me right now. But by the title I think Mr. Khan was proving this for all polygons. For all convex polygons, the sum of the exterior angles is always 360 degrees.
Could you find that concave angle by making it into a triangle?
I have a question please answer...does this rule of 360 degrees apply to all polygons?
Yes. It does. Now when you start thinking about regular polygons, really cool patterns start to show up.
wouldn't an un closed polygon have an exterior angle something different than 360
Is the sum of exterior angles 360 for all polygons? And how about triangles?
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!
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:
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!
So are all exteriors just 360 and then you divide the shape for interiors with triangles?
at 1:14 how am i supposed to copy the angle without a protractor? and if i had a protractor, i could easily use it to find the angles without doing this
I am still a little confused. What are convex and concave polygons?
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?
Anything that goes around that doesn't have a beginning or end = 360 degrees.
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.
In the video, Sal also did not know about the measurement of any triangle, but he was able to prove it anyways.
I know that the sum of the measures of the exterior angles of a polygon always equals 360 degrees. Is this true for concave polygons? If so why not, and how would you find the sum of the measures of the exterior angles of a concave polygon?
It is used to find sum of the interior angles. N = number of sides.
When you know the number of triangles formed,how do you find the sum of the interior angles of a polygon
You multiply the number of triangles by 180 and that's the sum of the interior angles.
I'm still confused as to how to find the sum of all interior angles. Can somebody please explain it in an easier way?
So is the exterior angles for every concave polygon then 360 degrees
but what if it is a regular polygon? do you just do the same thing, knowing that exterior angles are the same as the interior angle or is there an another way to find out.
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?
If the sum of interios angles of any polygon is (s-2)*180, then sum of all exterior angles must be 360.Considering that number of exterior angles equal the number of vertices of the polygon, then the sum of exterior angles must be equal to [[180*number of Vertices] - [s-2]*180 (i.e.the sum of the interior angles)].Why does this not work for concave polygons?
For the concave vertex there isn't an exterior angle in the traditional sense as it's going to be greater than 180 degrees. If you did try and calculate one anyway, you would have a negative angle, at which point the angles would still sum to 360.

This is a good site to see how it works.
The most important thing to keep in mind is that in case of convex polygons every *internal* angle's measure is less than or equal 180 degrees and concave polygons will *always* have at least one interior angle with a measure that is greater than 180 degrees.
What is the difference between concave and convex polygons? (P.s.: did I use the right terms?)
Concave polygons are polygons that "cave in". If you extend all the sides forever, at least two lines will go through the polygon. Convex polygons are polygons which don't "cave in". If you extend all the sides forever, none of the sides must go through the polygon.
Hope this helps! (P.s: You did!)
For all polygons would the sum of the exterior angles be 360 degrees or would it be 90 degrees for each exterior angle?
can you make a video with how to find the measure of the sides of the parallelograms and rhombus and how to find the angles using their properties
is the polygon , means more than one sides , am a little confused??
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).
How would you work out the exterior or interior angle of a concave polygon?
take the interior angle next to and subtract it from 180 or take the 2 opposite interior angle from the exterior angle and add them up
I'm still terribly confused as to when to use the triangle concept and when to use the circle concept. Sometimes it's hard to tell the difference between concave and convex polygons because the concave polygons are not always "caved" in.
Polygons are composed of straight sides. A circle does not have straight sides, it has curving sides
so a concave polygon has one angle greater than 180 degrees?
Yes, and for any convex polygon.
1 Vote