We're going to learn how to make a sphere out of insulation foam (extruded polystyrene). What you do with that sphere is your own business.

Step 1: Materials 'n Such

You're going to need:

Insulation foam - found at you local home supply store. I used 4'x8'x2" sheets. around 4 of them for this sphere. (sorry i don't know exactly how many, i was also working on another project simultaneously

Spray Adhesive - Make sure it's compatible with foam. I recommend Loctite High Performance Spray Adhesive , that stuff worked great, I was surprised how well it worked.

Cutting Tools - Sure, you could use a serrated bread knife, in fact that will come in handy later, but for the major cutting, do yourself a favour and buy a hand Jigsaw! I spent way too much effort cutting out large circles with a knife when I realized that tools had been invented to make life easier, and it did. My jigsaw cost about 30 bucks, and it was so worth it.

Large Chalkboard Compass - I found mine at a school supplies shop. this was a real life-saver. I first tried a string on a nail, but it was inaccurate and a real pain to use. The large compass made drawing large circles a breeze. I know, imagine that, tools made for our convenience.

Shaving/Sanding tools - get a hand rasp like the one pictured, It was the most efficient and least blister inducing. also get a couple different grades of sand-paper, coarse and fine. Use smaller electric sanders if you like.

Measuring Tape

Sharpies - Sharpies are the best.

Tunes! - get some good tunes goin! ask for me for recommendations, go ahead.

Step 2: Measuring Circles

Our sphere is going to be made of stacked circles, measured accordingly, so when the edges are sanded off, the result is a (nearly) perfect sphere.
This part gave me trouble, I couldn't figure out how to measure the circles so they descended in diameter the proper amount. I didn't want to end up with an oblong shape, and I didn't trust myself to eyeball it. I'm sure that to most of you fond of geometry, this was obvious, but it worried me until the solution came to me in an early morning epiphany.
So...
Drawa circle with the diameter of the sphere you want to end up with. this first circle you're probably only going to use for measuring purposes.

Drawa line through the center, marking the diameter

Drawparallel lines down to the edge of the circle, 2 inches apart. (since the foam you are working with is 2" wide, if it's not then adjust accordingly)

Now what you essentially have is a cross-section of the sphere. You now know how many circles you need, and the diameter of each circle. so each 2" wide section is the length, or diameter, of a circle you need to cut from the foam sheets. You'll need to cut two of each (unless you want a hemisphere, maybe that's what you want.) also, make sure to measure from the longer or wider line of each cross-section.

Step 3: Draw and Cut the Circles

Measure each circle from your sphere cross-section. In fact, just measure the radius of each circle on your cross-section directly with the compass, that way there is no measuring tape needed from here on out. draw them out on the sheets using your compass. (attach a sharpie onto the compass instead of chalk).

some tips:
number each circle when you draw it out, so you can easily keep track of their stacking order.

if you don't need your sphere to be solid (insulation foam is surprisingly sturdy) you may want to cut smaller circles out from the center of larger ones to conserve foam. (although, if you plan on drilling a hole through the already marked centers of each circle to stack them straight using a wooden dowel or something to align them, then maybe you want to keep the centers, it's an idea. I just eye-balled the alignment when it came to it)

Step 4: Stack and Glue the Circles

Stack the circles in the number you numbered them, or in descending diameter. glue them together with the spray adhesive. The tricky part is aligning the center of each circle correctly. I just eyeballed it, because the centers of most of my circles where removed. One thing I thought of that might help, is:
drill a hole in the center of each circle (using a drill that has some kind of guide to keep the hole straight and perpendicular, (I didn't have such a drill) then push a straight dowel or rod through the center of all the circles to keep them aligned by the center)

Step 5: Cut, Rasp, and Sand

Yep, that's about it.
As cool as it looks already, you probably want to get rid of the edges all around the surface.
So use your hand rasp to scrape of those edges. This is where the serrated knife might also come in handy. It can be nice to use to remove some of the larger angles. I find in this whole process, it's nice to have a couple of tools to switch it up. Use the sandpaper to smooth it out. When the edges are all smoothed down, you have a nice smooth Sphere! now make it into a Globe, or a Death Star, or a big non-bouncy beach ball, a rolling stone of death (indiana jones style), maybe an eyeball, someone please make a giant eyeball. Just make sure the paints you use are okay for foam. I think most water based or latex paints are good. a lot of spray paints - bad.
Enjoy your sphere. Also have fun cleaning up that mess, cause it's a big mess. You're going to need a good vacuum cleaner.