## Main Question or Discussion Point

Hi, guys. In a previous post this problem was given: 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. How do you attack this problem? I'm not sure where to start or how to relate L,W, and H to anything. Thanks for your input.

first picture the cube with length l, width w, and height h. Now, everypoint that lies within 1 of this cube becomes the new shape that we need to find a volume for. I started by breaking down the new cube into the old cube pluse 6 new sides, 12 new edges, and 4 new corners. Lets start with the sides. Each side is 1 unit wide, long or high, depending on what side you decide to start with. so you add (*=times)2*l*w+2*l*h+2*w*h to the volume of the original cube. Since there are six sides, you must add each side twice, hence 2*. Now you have the original cube plus the sides of your new shape. Next, we move on to the edges, l,w, and h. The edge of the new shape will look like 1/4 of a cylinder, or, 1/4 the area of a circle *l,w, or h. Since the area of a circle is pi*r^2, the volume of each new edge type shape is pi/4*l,w, or h, depending. Since there are 4 edges with l, 4 with w, and 4 for h, we can cancel the denominator to come up with pi*l+pi*w+pi*h. Finally, the corners. pasted together they constitute a unit-sphere, so there is an extra contribution of 4/3 pi to the volume of S:
lwh+2lw+2lh+2hw+pi(h)+ pi(l)+pi(w)+4/3Pi

EnumaElish
I promise to write again if I can think of anything even brighter. 