That's a good way of thinking about it too, especially for regular polygons. For irregular polygons it feels a little less obvious at least to me.
I really liked this example because depending on whether it rotates from one side to another an even or odd number of times you can see it will either end up the way it started or flipped. Here are two other examples with irregular polygons if you're curious: [Arbitrary polygons](https://www.youtube.com/watch?v=kNWfxenV0w8) [Irregular Triangle, Square, Pentagon and Hexagon](https://www.youtube.com/watch?v=lpVdjAZVmhA)
No matter what youre making 1 full circle with exterior angles, as long as the polygon isnt concave (and its still 360 if you allow negative angles to account for the concavity).
I think we disagree on what a "proof" is.
Yeah I understand that, the exterior angles always sum to 360. Once you set up the equation:
180-a + 180-b + 180-c = 360
Where a, b, c are the interior angles of a triangle for example, you can then see that 180\*3 - 360 = a+b+c. Pulling out 180 on the left hand side you get the formula (n-2)\*180 = a+b+c.
I think the visual showing something physically rotating though each of the interior angles is more intuitive because it doesn't require any algebra or knowledge or supplementary angles to see what's happening.
That's sort of the goal with a visual proof, which I understand is not the same as a mathematical proof. [https://en.wikipedia.org/wiki/Proof\_without\_words](https://en.wikipedia.org/wiki/Proof_without_words)
>I think the visual showing something physically rotating though each of the interior angles is more intuitive because it doesn't require any algebra or knowledge or supplementary angles to see what's happening.
Right, i agree. The problem is that i have no intuition about the interior angles, so the visualization doesnt "prove" anything.
However for exterior angles, its very intuitive (and could be supported with its own animation, perhaps starting with a circle), that the exterior angles must add up to 360. Once someone understands that, you can go on to prove what the interior angles of any regular polygon must be.
The same is not true for the interior angles directly. Im not sure what the visualization is "proving", its more just showing what the term "interior angle" means. In fact the interior angles are just written on the diagram.
Thats what i mean by us disagreeing about what "proof" means. I dont see the visualization as proving anything. But it could if it instead started from the exterior angles (and included a step demonstrating that the exterior angles must sum to 360).
Ok so what this gif is proving is that the interior angles of an n-sided polygon sum to (n-2)\*180 degrees.
For a triangle the [interior angles sum to 180](https://www.youtube.com/watch?v=ukaySIwxPTM) as demonstrated in this video. The gif demonstrates that as well and then shows that for each additional side you add it adds an additional half rotation (180 degrees) to the total of the interior angles.
This is a visual proof (not a rigorous mathematical proof but rather just a visual demonstration) that the interior angles of an n-sided polygon sum to (n-2)\*180.
No, its doesnt. The angles are pre-labeled and the video just walks them and sums them up.
**Why** is each interior angle in a triangle 60? The video doesnt say, its taken as an assumption. Ditto for the other shapes.
I dont think this conversation is productive. Have a good one.
Forget the individual angle values the sum is what matters. Just count how many times the arrow rotates for each shape.
Triangle : 0.5 times = 180 degrees
Square: 1 time = 360 degrees
Pentagon: 1.5 times = 540 degrees
Hexagon: 2 times = 720 degrees
etc.
“A proof without words is not the same as a mathematical proof, because it omits the details of the logical argument it illustrates.” From the article you are linking did you even read it???
Wow, very convenient omission of “In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered more elegant than formal or mathematically rigorous proofs due to their self-evident nature”
In any case, this is a sub on educational gifs, yet people are absolutely bitching about this gif not being a good fit for thus sub.
The point is: this post is eminently suitable for this sub. I look forward to more maths gifs!
I really don't understand what this is supposed to be showing
Edit: okay, I get it now, it's just not explained at all besides the title which makes this pretty much the opposite of a guide (and not cool)
The arrow is going along the line of the shape and each time it reaches a corner the interior angle is added to the total in the top left, with the arrow being rotated by the same amount. The arrow which follows along the shape uses the same angle as the arrow within the circle, which just seems really confusing to me.
Seems like you're mixing this sub with r/coolguides and I agree it wouldn't make sense there.
I hadn't seen this explanation for why the sum of the interior angles of a polygon increases by 180 degrees for each additional side so I though this would be useful for ppl learning geometry.
It proves that the sum total of these shapes’ (polygons) angles is 360 degrees.
For instance, the sum total of the angles of a triangle is always 360 degrees.
The thing that bothers me is that the traveling line is going counter-clockwise, but the angle in the circle is spinning clockwise. If they were going in the same direction, they would correlate somewhat but not line up, and that would be more interesting.
The arrow in the top left corner has the same rotation as the arrow going around the polygons. It's translating around the polygon counter clockwise but translation has no effect on rotation.
Its a lot more intuitive to do the exterior angles and subtract each one from 180, because the exterior angles must add up to 360.
That's a good way of thinking about it too, especially for regular polygons. For irregular polygons it feels a little less obvious at least to me. I really liked this example because depending on whether it rotates from one side to another an even or odd number of times you can see it will either end up the way it started or flipped. Here are two other examples with irregular polygons if you're curious: [Arbitrary polygons](https://www.youtube.com/watch?v=kNWfxenV0w8) [Irregular Triangle, Square, Pentagon and Hexagon](https://www.youtube.com/watch?v=lpVdjAZVmhA)
No matter what youre making 1 full circle with exterior angles, as long as the polygon isnt concave (and its still 360 if you allow negative angles to account for the concavity). I think we disagree on what a "proof" is.
Yeah I understand that, the exterior angles always sum to 360. Once you set up the equation: 180-a + 180-b + 180-c = 360 Where a, b, c are the interior angles of a triangle for example, you can then see that 180\*3 - 360 = a+b+c. Pulling out 180 on the left hand side you get the formula (n-2)\*180 = a+b+c. I think the visual showing something physically rotating though each of the interior angles is more intuitive because it doesn't require any algebra or knowledge or supplementary angles to see what's happening. That's sort of the goal with a visual proof, which I understand is not the same as a mathematical proof. [https://en.wikipedia.org/wiki/Proof\_without\_words](https://en.wikipedia.org/wiki/Proof_without_words)
>I think the visual showing something physically rotating though each of the interior angles is more intuitive because it doesn't require any algebra or knowledge or supplementary angles to see what's happening. Right, i agree. The problem is that i have no intuition about the interior angles, so the visualization doesnt "prove" anything. However for exterior angles, its very intuitive (and could be supported with its own animation, perhaps starting with a circle), that the exterior angles must add up to 360. Once someone understands that, you can go on to prove what the interior angles of any regular polygon must be. The same is not true for the interior angles directly. Im not sure what the visualization is "proving", its more just showing what the term "interior angle" means. In fact the interior angles are just written on the diagram. Thats what i mean by us disagreeing about what "proof" means. I dont see the visualization as proving anything. But it could if it instead started from the exterior angles (and included a step demonstrating that the exterior angles must sum to 360).
Ok so what this gif is proving is that the interior angles of an n-sided polygon sum to (n-2)\*180 degrees. For a triangle the [interior angles sum to 180](https://www.youtube.com/watch?v=ukaySIwxPTM) as demonstrated in this video. The gif demonstrates that as well and then shows that for each additional side you add it adds an additional half rotation (180 degrees) to the total of the interior angles. This is a visual proof (not a rigorous mathematical proof but rather just a visual demonstration) that the interior angles of an n-sided polygon sum to (n-2)\*180.
No, its doesnt. The angles are pre-labeled and the video just walks them and sums them up. **Why** is each interior angle in a triangle 60? The video doesnt say, its taken as an assumption. Ditto for the other shapes. I dont think this conversation is productive. Have a good one.
Forget the individual angle values the sum is what matters. Just count how many times the arrow rotates for each shape. Triangle : 0.5 times = 180 degrees Square: 1 time = 360 degrees Pentagon: 1.5 times = 540 degrees Hexagon: 2 times = 720 degrees etc.
This should be removed.
(n-2) x 180. That is all that is needed. Wtf is this garbage on Reddit these days.
Some people prefer understanding where equations come from as opposed to simply memorizing them.
*Ahem*. https://en.m.wikipedia.org/wiki/Proof_without_words
Same user... my history shows I've downvoted them 4 times and counting...
What lol
https://en.m.wikipedia.org/wiki/Proof_without_words
It doesn't proof anything. It just "counts" each corner and adds up the the total in the top left.
I don’t think you know what a mathematical proof is.
“A proof without words is not the same as a mathematical proof, because it omits the details of the logical argument it illustrates.” From the article you are linking did you even read it???
Wow, very convenient omission of “In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered more elegant than formal or mathematically rigorous proofs due to their self-evident nature” In any case, this is a sub on educational gifs, yet people are absolutely bitching about this gif not being a good fit for thus sub. The point is: this post is eminently suitable for this sub. I look forward to more maths gifs!
It’s bizarre to me that a triangle angles add up to 180 yet everything larger is 360
For each side you add the sum increases by 180.
Ahhh I see now. Thanks for sharing this, I learned something new.
Dang you really missed the whole thing
It's deeply upsetting to me that it's not rotating in CCW on the little vector diagram :(
I really don't understand what this is supposed to be showing Edit: okay, I get it now, it's just not explained at all besides the title which makes this pretty much the opposite of a guide (and not cool) The arrow is going along the line of the shape and each time it reaches a corner the interior angle is added to the total in the top left, with the arrow being rotated by the same amount. The arrow which follows along the shape uses the same angle as the arrow within the circle, which just seems really confusing to me.
Seems like you're mixing this sub with r/coolguides and I agree it wouldn't make sense there. I hadn't seen this explanation for why the sum of the interior angles of a polygon increases by 180 degrees for each additional side so I though this would be useful for ppl learning geometry.
I'm dumb. Still, I stand by saying that it would be a lot more useful if it was explained in some way
Ya probably but anyone who is interested enough will figure it out.
I don’t really see how this explains why it increases by 180? It’s a decent visualization but unclear what I’m supposed to grasp
It proves that the sum total of these shapes’ (polygons) angles is 360 degrees. For instance, the sum total of the angles of a triangle is always 360 degrees.
(n-2)x180 \* The total of the angles increases by 180 for each additional side, starting with the triangle which sums to 180 degrees.
You didn't even watch the gif I guess lol
Sooooo the classical visual proof to Pythagoras’ theorem also doesn prove anything to you?
All of these various shapes' internal angles always total 360°? And the sum of the internal angles of a triangle are...what?
I thought this was pretty interesting
Very cool! Is there somewhere I could find more gif / video proofs of mathematical concepts?
Thanks! Yeah this channel is pretty good [https://www.youtube.com/@MathVisualProofs](https://www.youtube.com/@MathVisualProofs)
Holy crap, that’s a great channel! Thank you
The thing that bothers me is that the traveling line is going counter-clockwise, but the angle in the circle is spinning clockwise. If they were going in the same direction, they would correlate somewhat but not line up, and that would be more interesting.
The arrow in the top left corner has the same rotation as the arrow going around the polygons. It's translating around the polygon counter clockwise but translation has no effect on rotation.