Need to find the volume of a rounded cube.

[alf_garnett]

Hello everyone. I'm currently working on a project (independent study and not homework) and I find myself having to calculate the volume of a cube with rounded corners and I'm stumped. Any help would be really appreciated.

The cube itself is 8 cubic meters (2m on each side) and the rounded corners are 406mm from what would otherwise be a corner point to the end of the curve in all three directions spreading out from the would-be corner point. I need to be fairly precise with this calculation but I'm afraid my mathematics know-how isn't getting me very far. I appreciate the fact that this a learning forum and so I do not expect a direct answer to this but if somebody would be good enough to provide me with a formula into which I can put these figures I would really be very grateful. I've included an image of my problem in case I'm not being very well understood. Thank you all for reading.

1. Find the volume of a cube with rounded corners
Volume of un-rounded cube 2³ = 8 cubic meters.
Length of rounding = 406mm

2. Width * Height * Depth ... after that I just dunno.



3. Completely Stumped.

 

[n.karthick]

Is the curve describing your rounded corner a circle?

 

[alf_garnett]

It is, yes. Sorry, I should have mentioned that.

 

[LCKurtz]

If you took all 8 rounded corners of the cube and put them together you would have a sphere of radius 406 mm wouldn't you? And that sphere plus the part that was shaved off would be a little cube, right? So you can figure out how much was shaved off...

 

[alf_garnett]

Ahhhh - how simple it all seems when it's spelled out :D lol - I feel so thick. Thank you very very much LCKurtz. I really appreciate that. I do believe I can work this out now. I'll get back and post my result soon. Thank you.

 

[alf_garnett]

So I came up with the answer 7.755 square meters (rounded off). I calculated the volume of the sphere of radius 406mm, subtracted that from the volume of a 812mm cube and then subtracted the result of that from my 8 square meters to get the answer above. 7.754941. I have to ask ... did I follow the right process?

 

[LCKurtz]

So I came up with the answer 7.755 square meters (rounded off). I calculated the volume of the sphere of radius 406mm, subtracted that from the volume of a 812mm cube and then subtracted the result of that from my 8 square meters to get the answer above. 7.754941. I have to ask ... did I follow the right process?

I didn't check your numbers but the process looks OK. Be sure about your units -- you have mm and m and volume is m³.