Chapter 3: Understanding quadrilaterals
Sum of the measures of the exterior angles of a polygon
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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
Discussion and questions for this video
 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 in particular exterior angles of the hexagon so that
 this angle equaled A, this angle B, C, D and E. 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 angle. So it was a bit of an involved process. But I want to show
 you in this video that 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 polygon (if you're picking these particular exterior angles I should say) and so the way to
 think about it is you can just redraw the angles. So lets just draw each of them, so let me draw this
 angle right over here, we'll call it angle A or the measure of this angle's A, either way let me draw
 right over here. So this going to be a convex angle right over here
 it's going to have a measure of A, now let me draw angle B, angle B, and i going to draw
 adjacent to angle A, and what you could do is just to think about it
 maybe if we draw a line over here, if we draw a line over here that is parallel to this line
 then the measure over here would also be B,because this is obviously a straight line,
 it would be like transversal, this of course a responding angles, so if u want to draw
 adjacent angle, the adjacent to A, do it like that, or whatever angle this is the measure of B
 and now it is adjacent to A, now let's draw the same thing to C
 We can draw a parallel line to that right over here. And this angle would also be C
 and if we want it to be adjacent to that, we could draw it there, so that angle is C
 C would look something like this, like that then we can move on to D, once again
 we do it in different color, you could do D, right over here or you could shift it over here
 it'll look like that, or shift over here, it'll look like that
 If we just kept thinking of parallel, if all of this line were parallel to each other
 So, let's just draw D like this, so this line is going to parallel to that line
 Finally, you have E, and again u can draw a line that is parallel to this
 right over here and this right over here would be angle E
 or you could draw right over here, 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 going all the way around the circle, either way
 it could be going clockwise or it could be counter clockwise but it will going all the way
 around the circle.
 And some of this angle, A+B+C+D+E is just going to be 360 degree
 And this is work for any convex polygon, and when I say convex polygon I mean it is not that dented words
 Just to be clear what I'm talking about, it would work for any convex polygon that is kind of
 I don't want to say regular, regular means it has the same size and angle,
 but it is not dented, this is a convex polygon. This right here is a concave polygon
 Let me draw this, right this way, so this would be a concave polygon
 Let me draw as it having the same number of side, So i just going to dent this two sides
 right very here. Is it right? Let me do the same number sides, So i do that, that, that, that
 and then that's the same side over there, Let me do that and then like that.
 This has 1,2,3,4,5,6, sides and this has 1,2,3,4,5,6 sides. This is concave, sorry
 this is a convex polygon, this is concave polygon, All you have to remember is kind of cave in words
 And so, what we just did is applied to any exterior angle of any convex polygon. I
 Am a bit confused. This applied to any convex polygon and once again if you take this angle
 and added to this angle and added to this angle, this angle, that angle and that angle and I'm not applying that all
 It's going to be the same and I just drew it in that way I could show u
 that they are different angles, i could say that one green, and that one some other colour
 they can all be different but if you shift the angle like this you can see that
 they just go round the circle. So, once again, I'll just add up to 360 degrees
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?

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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.........
Hope this will help you out of this!
Hope this will help you out of this!
Yes, it is always!
Imagine punching a side in on a polygon to form a concave polygon, one with an inverted side, from a convex polygon.
No, only for convex polygon. Here's how you figure it out: draw a line segment starting and ending inside the polygon. If the line crosses over any space not filled by the polygon it's concave. Other wise it's convex.
convex polygons... and triangles (i like triangles!)
Imagine that a polygon is nothing more than a incomplete circle.
Yes the sum of the exterior angles of a CONVEX POLYGON equals 360 degrees
Yes that whats my dad said
Yes, the sum of all exterior angels of a convex polygon is 360 deg, And what's the logic behind this is explained in the video.
"Sum of the exterior angles of CONVEX polygon:"
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A basic way of finding the measure of interior angles of a convex polygon: Multiply 180 by the number of sides of the polygon, then subtract 360
Yes, but only for convex polygons
The sum of all external angles on a polygon = 360 degrees. Kind of like in triangles all interior angles = 180 degrees. ;)
no. different polygons have different sides . that's according to no. of sides they have.
yes it is,as u can see on video
It only works for convex polygons since if you think about it, a concave polygon doesn't exactly have exterior angles.
JUST CONVEX POLYGONS!
for all convex polygons
yes
yes
Yes.
yes for all pOlygons lol
yes
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?
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.
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.
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
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
because just is that true
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
The measure of an interior angle plus the measure of its exterior angle is equal to 180 degrees. So if you have a polygon that has in angle that is 200 degrees, what is it's exterior angle? Well let's see, 200 plus what is equal to 180? 20! But negative angles aren't possible...
Using negative exterior angles might get the sum to 360 in some cases, but don't rely on them  they defy logic!
Using negative exterior angles might get the sum to 360 in some cases, but don't rely on them  they defy logic!
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What! There are TWO reflex angles for conCAVE polygons.
Can you have a negative angle?
Yes, if you say that angles measured counterclockwise are positive and those measured clockwise are negative : )
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No, angles are defined as being between 0 and 360.
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.
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if n is a number of sides, n + 1 cannot be a shape
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.
that would better be stated as it is going inward at multiple point along its perimeter, reinforcing its concavity rather than negating it.
No a star is concave. The space between two adjacent arms caves inward.
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Wait a minute... A star is both right? Because it is going inward and outward right?
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 circumcircle of a polygon equal a circle? If so, is a circle a polygon?
Stated another way 
Since there is no theoretical limit to a polygon at what point does the circumcircle 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 ).
An infinite number of sides will *not* make a curve. It only appears to make a curve due to our ability to differentiate things. However, if you are able to look at the sides at the proper scale there would be no question about whether you were looking at a figure that had sides.
Think about viewing things with an electron microscope. They look very different from when you view them with the naked eye. Moving toward the other end of the spectrum, think about how we perceive things when we stand on the earth; it seems "flat" to us when we know that it is actually curved.
In the world of mathematics there is the ability to get infinitely small or infinitely large. So, it is possible to view things at whatever scale is needed to be able to detect the difference.
A circle is not a polygon and never will be. A circle is the collection of all points that are equidistant from a central point. A polygon will never meet that definition.
Think about viewing things with an electron microscope. They look very different from when you view them with the naked eye. Moving toward the other end of the spectrum, think about how we perceive things when we stand on the earth; it seems "flat" to us when we know that it is actually curved.
In the world of mathematics there is the ability to get infinitely small or infinitely large. So, it is possible to view things at whatever scale is needed to be able to detect the difference.
A circle is not a polygon and never will be. A circle is the collection of all points that are equidistant from a central point. A polygon will never meet that definition.
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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?
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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yes all polygons have an external measure of 360 degrees
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.
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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a concave polygon goes in on itself but a convex polygon doesn't
A convex polygon is a polygon with no angles that have a measure greater than or equal to 180.
What is the meaning of anticlockwise?
it is the same as counterclockwise, 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.
Or if you start at the top of a circle, and go down and around to the left.
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At 3:10 Sal said something like clockwise and counterclockwise..
What does he mean by clockwise and counterclockwise ?
What does he mean by clockwise and counterclockwise ?
Clockwise on the clock rotation goes as toprightdownleft.
Counterclockwise on the clock rotation goes as topleftdownright.
Most clocks go Clockwise.
Counterclockwise on the clock rotation goes as topleftdownright.
Most clocks go Clockwise.
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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.
From now on, I think I'll wait until the end of the video before commenting.
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always wait until the end of the video to ask a question.
yeah man you should wait
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.
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Can a polygon have a higher angle than 360?
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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?
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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)
Interior Angle = π  exterior angle (in radians)
Interior Angle = 180°  exterior angle (in degrees)
What is the difference between convex and concave shapes ?
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"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.
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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Yup, haven't seen a "mathematically rigorous" proof either, that's why I was asking. Wasn't sure there is one. :) But I'm guessing I'll find it in some video somewhere as I progress further.
Thanks for the answer!
Thanks for the answer!
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, stepbystep, 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.
So, I'd agree it is annoying at first, but it is a skill worth knowing.
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Yes, when I was in High School we had to do it too. And I agree they are very annoying, but they do lead to a better understanding of geometry.
then how do you find the interior angles of a convex polygon?
http://www.khanacademy.org/math/geometry/polygonsquadsparallelograms/v/sumofinterioranglesofapolygo this is the video. hope that helps! :)
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Watch the video "Sum of the interior angles of a polygon".
we all know that!
To find out only one exterior angle(of a convex polygon), do you do 360/ number of sides?
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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.
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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?
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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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!
Hope this helps!
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What is the sum of the exterior angles of a concave polygon?
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Why isn't an exterior angle the angle that goes all the way around the corner, greater than 180?
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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So confusing not helpful
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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!
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.
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I do not know what ur talking about, but I know that a property of a parralelagram is that the diagonals congruently bisect each other.
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.
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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.
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.
Can't you just "pop out" concave angles to make a convex polygon? The change of the angle would be the same, right?
No, because the two angles formed by the caved in part of a concave polygon wouldn't be there anymore, and the convex polygon wouldn't be congruent to the concave polygon. I hope this makes sense! :D
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 ?
Of course he's not talking about exterior because we are watching that right now.
He's talking about Quadrilateral Overview.
He's talking about exterior angles, not interior angles.
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.
HELP! The exercises *and* the video aren't very explanatory and I' m confused in the exercise "Angles of a polygons".
No matter what type of polygon you have, the sum of the exterior angles is ALWAYS equal to 360°. If you are working with a regular polygon, you can determine the size of EACH exterior angle by simply dividing the sum, 360, by the number of angles. Remember, this will ONLY work in a regular polygon (CONVEX).
http://www.regentsprep.org/regents/math/geometry/gg3/lpoly3.htm
http://www.regentsprep.org/regents/math/geometry/gg3/lpoly3.htm
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
wouldn't an un closed polygon have an exterior angle something different than 360
a "unclosed polygon" doesn't exsist
A polygon is a flat shape consisting of straight lines that are joined to form a closed chain or circuit. Therefore, there is no such thing as an unclosed polygon.
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 nonrigorous 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
More at Wikipedia: http://en.wikipedia.org/wiki/Convex_polygon
A polygon in which all the interior angles are less than 180 degrees.
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
How do you find the sum of the exterior angles of a concave polygon?
They still add up to 180˚ provided that you remember to add up the "negative" angles properly.
I can't find the video mentioned in this one about finding exterior angles the other way. Only the sum of interior angles of a polygon video shows up under the "need more help? Watch this video" area and I can't seem to find the video on the side bar thingy (don't know what to call it) The angles of a polygon practice includes exterior angles and that's the practice I'm working on right now. Please help! Thank you.
At 1:39,shouldn`t 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.
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The sum of the exterior angles of any convex polygon is 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?
A concave surface curves inward. The word is easy to remember because a concave indentation in a wall makes a cave. A convex surface curves outward. Like many pairs of antonyms that are relatively rare and similar in sound, these two adjectives are easy to confuse.
Concave and convex are also geometrical terms; a concave polygon has at least one angle greater than 180 degrees, and a convex polygon is made of angles each less than or equal to 180 degrees.
http://grammarist.com/usage/concaveconvex/
Concave and convex are also geometrical terms; a concave polygon has at least one angle greater than 180 degrees, and a convex polygon is made of angles each less than or equal to 180 degrees.
http://grammarist.com/usage/concaveconvex/
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.
No because it has those extra areas outside the circle. This is a really great question though.
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.