Angles and intersecting lines

# Angles with triangles and polygons

## Sum of the exterior angles of convex polygon

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

## Sum of the exterior angles of convex polygon

Discussion and questions for this video
Is 360 degrees for all polygons ?
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.........

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?
1 Comment

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?
Actually, it is 360°, according to http://en.wikipedia.org/wiki/Exterior_angle
Comment on A's question
Can you have a negative angle?
1 Comment
Yes, if you say that angles measured counterclockwise are positive and those measured clockwise are negative : )
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 ).
1 Comment
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?
Yes, the answer is 360 degrees and will always be 360 degrees for any convex polygon. Good job on figuring that out!
is a star considered as a convex polygon?
No, it is concave because it has an angle greater than 180 degrees( also known as a reflex angle.
what is a convex polygon?
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.
What is the meaning of anticlockwise?
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.
1 Comment
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.
Why didn't Sal define convex?
3:29 Oh wait, never mind...

From now on, I think I'll wait until the end of the video before commenting.
1 Comment
What does he mean by clockwise and counterclockwise ?
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?
1 Comment
Yes you can, but I don't think you would need an angle more than 360 degrees.
1 Comment
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)
1 Comment
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.
then how do you find the interior angles of a convex polygon?
To find out only one exterior angle(of a convex polygon), do you do 360/ number of sides?
No you do not.
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?
What is the sum of the exterior angles of a concave polygon?
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.
Are concave polygons also add up to 360 degrees?
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!
explain how you can determine if a polygon is concave
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.
is A+B+C+D+E+G+H+I= 360 or is it something else?
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.
1 Comment
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.
How do you find the exterior angles of a conCAVE polgon, then?
What previous video is he referencing in the beginning?
Sum of Interior Angles of a Polygon ?
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.
1 Comment
Would students benefit from learning that the sum of exterior angles is tau radians?
what if there is more than six angles?
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.
At 3:29, doesnt convex have a second meaning?
Could you find that concave angle by making it into a triangle?
how do you determine how to solve a regular polygon equation
what do you mean by solving to find a regular polygon equation?
convex means what exactly?
What is a convex region
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
a "unclosed polygon" doesn't exsist
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!
How do you find the perimeter of Convex and Concave polygons?
what is a convex polygon
?
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
does a concave polygon also have a measure of 60 degrees?
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?
what is the exact equation for exterior angles
add all the measures of the exterior angles and set them equal to 360
At 1:39,shouldnt angle A be + to point B?
Is this true for all polygons that the exterior angles are equal to 360?
Yes. All exterior angles add up to 360.
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?
wait so why cant it work for 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.
how do i algabraically make a hexagon add up to 360?
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.
What about the formula: (n-2) x 180?
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?
Are all exterior polygons 360 degrees?
Yes, the sum of the exterior angles for ANY polygon is 360, making the very interesting formula:
S(Exterior Angles) = 360 degrees`
What is a convex polygon?
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!
What makes a polygon convex?
So is the exterior angles for every concave polygon then 360 degrees
in the video the captions are in Chinese ....what do I do?
wait, so is every polygon's exterior angle is 360 degrees?
it's applicable only for CONVEX...don't forget that part!
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.
how did mathmatishins crieit degreases
p.s sorry for my spelling
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. http://www.mathopenref.com/polygonexteriorangles.html
What is the difference between convex and concave polygons?
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!)
Why doesnt it work for a convave polygon?
does this apply to all polygons
If they are convex, yes
what is the sum of interior convex angles?
For all polygons would the sum of the exterior angles be 360 degrees or would it be 90 degrees for each exterior angle?
The exterior angles of any polygon equals 360 degrees.
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).