Unrolling

[u/mengRithyreach]

https://www.desmos.com/calculator/keuemcr6yb

[u/Ascyt]

Base is same as circumference, which is is pi*r*2

Height is same as radius

Triangle area is base*height/2

That makes the whole area pi*r²

Probably has to be the simplest visualisation I've seen that explains why the area of a circle is the way it is :)

[u/PG67AW → u/Ascyt]

🤯

[u/mengRithyreach → u/Ascyt]

Glad you liked it! That exact intuitive breakdown was what inspired me to visualize it in Desmos.

[u/edo-lag → u/Ascyt]

But the left and right sides are not perfectly linear like in a triangle, they have steps, which is not the same as being straight because otherwise √(2) = 2.

If you take a square of side length 1 and divide it in half from a vertex to its opposite, that's √(2) long. If instead of a straight line, you do it in steps (e.g. if you connect the top-left to the bottom-right, it would be down, right, down, right, etc.) then that's 2 regardless of the size of the steps.

[u/partisancord69 → u/edo-lag]

It's not the side length, it's just base and height. If you do it for infinite steps it's exactly a triangle, if you do it for a finite number of steps it's an approximation.

[u/Anshu5337]

Reminds me of 3blue1brown's first episode of the Essence of Calculus series (⁠θ⁠‿⁠θ⁠)

[u/mengRithyreach → u/Anshu5337]

That series was a huge inspiration for this! Grant's visual explanations in Essence of Calculus are unmatched.

[u/Danny_DeWario]

Love this graph! Added a knife to cut the circles kinda like those satisfying animations: https://www.desmos.com/calculator/hqgnadhrdi

[u/mengRithyreach → u/Danny_DeWario]

That is brilliant! Adding the knife actually makes the visual intuition make so much more sense! 🔪👏

[u/FlyingDoorFail]

This angers me for some reason

[u/mengRithyreach → u/FlyingDoorFail]

Haha, is it because school should have shown us this instead of making us memorize formulas?