If I had a sphere with a radius of 100 meters, a diameter of 200 meters, a volume of 4,188,790.20 square meters, and I wanted to place within this sphere a single dot (one dimensional so it doesn't take up any extra space and there is no displacement --if you're thinking in terms of water--), and I need to have one dot every 50 meters, what is the formula I would use to determine that? I thought it was just divide the volume by the number 50, but that comes out with a large number like 83,775.80, which seems insanely huge for something with just a diameter of 100 meters. What am I doing wrong here? This isn't a homework question, just something I'm trying to throw together for an experiment I'm doing in my personal time.
I presume you are trying to fill the volume of the sphere. You need to divide by 125000 (50^{3}) not 50. I got 33. Note volume is cubic meters, not square meters.
Since the "dots" are to be 50 m. apart, each dot could be thought of as the center of a sphere 25 m. in radius, so you need to divide the volume of the large sphere by (4/3)##\pi (25)^3##. This wouldn't give you the exact number of points inside the sphere, as it doesn't take into account how the small spheres are arranged inside the larger one. One of the areas of mathematics deals with sphere packing inside of geometric objects. Mathematicians who work in this area consider such simple examples as how oranges are stacked in a pyramidal pile on up to how spheres can be packed in much higher dimensions, which has application in the area of digital communications.