What is the Concept of Volume?

[batballbat]

i am having confusion with the concept of volume. Does a sphere with a hole have the same volume as the volume of a sphere? I think it doesent, but if the hole is sufficiently small, they would hold the same amount of fluid. Also the hollow space of sphere counts as its volume. Then why doesent a curved plane which covers some hollow space count as its volume. please give me a detailed definition of volume

 

[Number Nine]

I'm not entirely sure what you're talking about here. What do you mean by "why doesent a curved plane which covers some hollow space count as its volume?"

Part of the problem seems to be that you're using strange terminology. Generally, the term "sphere" refers refers to the surface of a ball, but you seem to be using "sphere" and "ball" interchangeably, and so I have no idea what a "hole in a sphere" is supposed to be. The notion of volume is an intuitive concept; if you want a "detailed" rigorous explanation, you need real analysis.

 

[batballbat]

a cylinder without a lid has the same volume as a cylinder with lid. So it is not clear to me. Same for sphere

 

[mtayab1994]

a cylinder without a lid has the same volume as a cylinder with lid. So it is not clear to me. Same for sphere

And you just said it. A sphere with a "hole" has the same volume as one that doesn't.

 

[HallsofIvy]

a cylinder without a lid has the same volume as a cylinder with lid. So it is not clear to me. Same for sphere

A cylindrical can, for example, does NOT have the same volume with and without the lid because these are actual physical things and both can and lid have volume.

But the mathematical, ideal, solid has boundary with 0 thickness. The "top" of such a cylinder has 0 volume.

 

[batballbat]

what is the volume of a sphere with a hole? so in geometry only bounded solids have volume, am i correct?

 

[mathman]

what is the volume of a sphere with a hole? so in geometry only bounded solids have volume, am i correct?

I think you need to be more precise in your descriptions. The region of 3 space defined by:

x² + y² + z² < r²

has the same volume as

x² + y² + z² ≤ r²

even though the first doesn't include the boundary, while the second does.