Volume of points within distance 1 from a box

[Yoda13]

Let B be a solid box with length L, width W, and height H. Let S be the set of all points that are a distance at most 1 from some point of B. Express the volume of S in terms of L,W, and H. Have fun.

 

[Timbuqtu]

Don't forget the corners: pasted together they constitute a unit-sphere, so there is an extra contribution of 4/3 pi to the volume of S:

$$L W H+2L W+2H W+2L H+\pi H+\pi L+\pi W + \frac{4}{3}\pi$$

 

[saltydog]

Ok thanks. Now I know what you guys were talking about. I though he meant a point INSIDE the box say (h,j,k). I think that's a little more difficult: what is the volume of the solid formed by all points inside the box that are a distance of 1 unit away from the point (say for a box 2x2x2)? Are there points in the box that yield a miniumum volume? Is it at the corners? So conceivably, we could construct a function of 3 variables: f(x,y,z), which yields this volume as a function of position in the box. Really, I think just calculating one such volume would be difficult.

 

[cmdro]

can someone please explain where the l(pi)+h(pi)+w(pi) came from. sorry I am a physics student and we were given a similar problem, and i just don't really understand. our was for a different shape and different distance but i think this helps.