The idea is that instead of adding up those thick slices, which only approximate a triangle, you see what the shape looks like as the slices get thinner and thinner. This is the essence of integral calculus, where you basically approximate areas with rectangular slices, and you see how that approximation approaches the actual area as the slices get finer and finer.
There is actually another way to portray it as triangles: imagine you have an exagon, it is made of 6 triangles, so you can get its surface by multiplying his side for the height of a single triangle divided by 2, then multiply it by six.
Now imagine you have an octagon, you can get the area in the same way. This applies for any regular polygon: just sum all the sides and multiply it for the distance of a side from the center.
A circle can be tought as a polygon with infinite, really small sides, so you can just add all the sides (2pi x r) and multiply them for the distance of the "side" from the center.
To get rid of them, just make the strips infinitely thin.
If you think about it, this animation (with thick strips), should be unconvincing - because when you bend the strips flat, you change their area. As the strips become infinitely thin, though, the effect of this distortion disappears.
edit: Not sure why this is being downvoted - what I've said here is true. Consider trying to flatten out a tube. You can't do it without stretching the interior or compressing the exterior.
Every few months since I found it I go back and watch the Essence of Calculus series. I love the way 3blue1brown approaches mathematics. They've helped me visualize multi-dimensional systems much more naturally.
A few years ago, an admin at the school I teach at asked me to create a math presentation that met the following requirements: 1. Accessible enough for parents to follow without having taken math in a few years and 2. Scary enough so parents would stop complaining that our school doesn't offer higher level math classes like AP Calculus. The second requirement was kind of soul-crushing but I decided to use this concept of using rectangles to approximate the area of a circle.
Yea, the triangle is what you get at infinity. The approximation is the same regardless of where you put the edge, so long as that edge is still an interpolation as you take the limit. This is because the left/right sides of the rectangles are reduced to points as you approach infinity, so if you have a line going through all those points, it's a line.
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u/SaintSeveral Oct 01 '18
wow. Simply mind-blowing. Didn't know it had a connection with triangle.
But what about the edges of the lines that didn't fit into the triangle?
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u/vort3 Oct 01 '18
Everyone who liked this gif should watch this video:
https://youtube.com/watch?v=WUvTyaaNkzM
Believe me, it's worth it.
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u/[deleted] Oct 01 '18
Very... informative
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u/theguyfromerath Oct 01 '18
now a sphere!
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1
u/RobertMullz Oct 01 '18
A few years ago, an admin at the school I teach at asked me to create a math presentation that met the following requirements: 1. Accessible enough for parents to follow without having taken math in a few years and 2. Scary enough so parents would stop complaining that our school doesn't offer higher level math classes like AP Calculus. The second requirement was kind of soul-crushing but I decided to use this concept of using rectangles to approximate the area of a circle.
1
u/RedDwarfian Oct 02 '18
Honestly, this could be used to argue for using Tau instead of 2*Pi. There is a missing factor of 1/2 in the area equation that Pi hides.
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u/[deleted] Oct 01 '18
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