One thing that still baffles my mind is, WHY exactly is it 1/3?? If you take a 2D cross-section of a cone within a cylinder, it's basically a rectangle with a triangle within it. You can bisect the triangle into two equivalent mirrored triangles and you get two rectangles that are halved diagonally. If you add all the resulting triangles together, you should get 2/4 or 1/2, not 1/3. Yes, it's not 2D but 3D, which is a whole other story, but if you circle this whole thing around, I still don't understand how you get 1/3, even though it intuitively makes sense.
I'm not sure but I'm pretty sure that we can test it, like try finding a funnel, and then a big cylinder like a water bottle, gallon jug of water etc. Put it inside, then figure out the volume of the cylinder, then the volume of the funnel. It won't be as accurate, but it will give you an idea of how it works. But still, like you said, it intuitively makes sense, though visually, it doesn't. Hope this helps!
I know right. I need some mathematical proof. So I made some up. I presume that the VOLUME of a cone is the AREA of a triangle times the length of the circumference of the circle created by the cone. sooooo AREA of a triangle (I'll call the base "r") = rh/2 circumference of circle = 2*pi*r when you multiple these together you get 2*pi*r*r*h/2 or h*pi*r^2 for some reason when I crunch the numbers I get TOO MUCH SPACE (???) hmm idk what to do.. So I pretend that the reason I get to much space is because I accounted for "r" twice in the equation OR because the amount of space I have is the volume of a 4 dimensional cone (I'm not sure if that true but....... ...) so I divide by 3 (because Length * Width * Height = the volume of a 3 dimensional object) and get the proper answer. h*pi*r^2 / 3 or 1/3 * h*pi*r^2 In other words... I'm in the same boat as you. I need mathematical proof.
its not a rigorous proof but u can think of how in 2d a triangle is half the area of a square with same height and width bc as the hypotenuse reaches the "peak" it loses half the area since its in 2 dimensions. Kind of the same happens with a cone but this time in 3d so as the side reaches the "peak" it loses 1/3 of the area. Hope this helps
Bro...if I didn’t come across this video in 10th grade I would have dropped out...the formula kept pissing me off. I ended up putting .3 and 1/3 in my calculator and kept giving me dumb answers. I finally put in a 3 and got it right. Phew...
becayse it doesnt staay constant throughout the calculation. In a cylinder, you can multiply the base times the height because the base stays the width throughout the whole shape. In this case, it gets smaller, so you cant have a constant multiplication through the triangle.
Exactly my question, the triangle is moving by the circumference therefore it's only logical that it's (triangle's area x circle's circumference) which would be piR^2h instead they just assume that it's 1/3 part of the cylinder 😬
