Alex Dishes Out Another Mathematical Quandary This one is about AREA OH NO

[galexcd 0]

My "circles don't exist" topic really started some fierce debate. Some people belived me, some people didn't, and some people got caught up in the whole .9~=1 thing. So just to cause more havoc I decided to post my theories on area! SO...Lets say I've got a two dimentional circle. Radius X. So the area of this two dimentional circle is piX^2. Alright so lets look at a cylinder. Lets say we take this circle and start turning it into a 3-dimensional object. We start pulling on either side and we make it into a cylinder. So mathamaticly I should be able to get the area of this cylinder by multiplying the area of the top by its height. Lets say the height is 4. If we take the original two dimensional cylinder and multiply it by 4, it would have no depth to it since each cylindar has a depth of 0 since they are all 2-dimensional. So that means all objects that have any depth at all have infinite area.So does this mean that every object, every thing we touch and see is actually infinitely big?

[t3jem 0]

Your looking at it as if the 3-dimensional object is made of 2 dimensional ones. In reality there is no "read" 2 dimensional object, even a drawing on a piece of paper has depth. So when you turn a 2D circle into a 3D cylinder you multiply by the hight because that is how you find the area of the cylinder. The problem of your confusion I think lies in the tool of measurment. When you define the Area of a circle you use "units ^2"; however, when definining the Volume of a cylinder you use "units ^3" so you can't really add the area to find a volume. Since area is 2D and Volume is 3D if you tried to find the volume of a 2D circle you would come up with 2 because there is no hight; however, when you add hight to the cylinder you are able to find the volume rather than the area. I hope this was at least somewhat coherent, I just got home from school and i'm never good at relaying mathematical ideas. I apologize, let me know if there's anything you need clarification on, I will be checking back.

[bbelnap2 0]

Your looking at it as if the 3-dimensional object is made of 2 dimensional ones. In reality there is no "read" 2 dimensional object, even a drawing on a piece of paper has depth. So when you turn a 2D circle into a 3D cylinder you multiply by the hight because that is how you find the area of the cylinder. The problem of your confusion I think lies in the tool of measurment. When you define the Area of a circle you use "units ^2"; however, when definining the Volume of a cylinder you use "units ^3" so you can't really add the area to find a volume. Since area is 2D and Volume is 3D if you tried to find the volume of a 2D circle you would come up with 2 because there is no hight; however, when you add hight to the cylinder you are able to find the volume rather than the area.
I hope this was at least somewhat coherent, I just got home from school and i'm never good at relaying mathematical ideas. I apologize, let me know if there's anything you need clarification on, I will be checking back.

I agree with t3jem. The problem is that you are trying to merge the 2 dimensional concept of area with the 3 dimensional concept of volume. They are not the same thing. A four inch tall cylinder, like a glass, can contain water. An infinite number of 2 dimensional circles will never hold anything.