cutting up cylinders...

take a diagonal slice across a cylinder, starting from the top 'rim', and finishing somewhere on the base. take the smaller of the two slices. given we know most of the dimensions (radius, height, angle of cut...) can we find the volume? i would guess so. but how?

Reply 1

Infinity_Kev

lol i had a silly idea this cant be right

treating it as a parralelogram

I took the larger piece but you could minus it from the original

beautifull.GIF5.0KB

Reply 2

SsEe

No. Just did a triple integral on mathematica. It's very very very not pretty!

Reply 3

Chewwy

Infinity_Kev

lol i had a silly idea this cant be right

treating it as a parralelogram

I took the larger piece but you could minus it from the original

i'm interested in the other kind of cut, which goes down through the base, not the side.

Reply 4

Chewwy

SsEe

No. Just did a triple integral on mathematica. It's very very very not pretty!

oh. how depressing.

Reply 5

BCHL85

can you creat a picture chewwy??

Reply 6

BCHL85

If you meant the cylinder was cut by a plane passing through A and E like in the image below, then here's the answer for it.
You said we knew r, h, angle of cutting, say x (x = EAC)

Then EC = a = h.tanx
V(cylinder ACEF) = ¼πa².h
Then you only need to divide that volume by 2, you'll get the volume of the small piece

edit: forgot 1/4 lol

Reply 7

Chewwy

BCHL85

If you meant the cylinder was cut by a plane passing through A and E like in the image below, then here's the answer for it.
You said we knew r, h, angle of cutting, say x (x = EAC)

Then EC = a = h.tanx
V(cylinder ACEF) = ¼πa².h
Then you only need to divide that volume by 2, you'll get the volume of the small piece

edit: forgot 1/4 lol

ACEF is not a cylinder.

Reply 8

BCHL85

chewwy

ACEF is not a cylinder.

why not then???
edit: this is the picture in 1 face ... plz imagine your head... I am not good at drawing

Reply 9

Chewwy

BCHL85

why not then???
edit: this is the picture in 1 face ... plz imagine your head... I am not good at drawing

get a glass of water, fill it less than 50%, and tilt. you'll see.

Reply 10

yazan_l

chewwy

get a glass of water, fill it less than 50%, and tilt. you'll see.

actually u cant see that ACEF isnt a cylinder using the glass of water... although it's a nice idea :biggrin:

attachment shows why it's not a cylinder... (it's a top view)

topview.PNG3.8KB

Reply 11

BCHL85

yazan_l

actually u cant see that ACEF isnt a cylinder using the glass of water... although it's a nice idea :biggrin:

attachment shows why it's not a cylinder... (it's a top view)

oh .. I made a mistake ... lol
very stupid of me. Let me try again, then

Reply 12

arktos

I've got an idea that it is the
(final cross-sectional area / initial cross sectional area) h
but thats probably quite stupid.

Reply 13

LeeC

I don't think you could do it with an integral (volume of revolution), could you? I mean any curve on a graph that I can think of would not revolve 360 degrees around the axis and give the desired shape. Hmmm....

Reply 14

Esquire

the volume would depend on the angle of the cut, the height and the radius. I'm not sure about integrals on mathematica, but this is definately a very complicated problem. It's calculus of 4 variables ie. f(x,y,z,w). pfft, ask a cambridge mathmo lol.