I have this shape which is half a sphere, the same shape top and bottom (see the other layer). I need to UV unwrap it so it projects naturally onto a blank image so I can paint it naturally (like if you took the shape as it is and simply unfolded it flat onto a table). I have tried and I don’t know how to get it unwrapped into its natural flat shape. It’s always distorted. Please help. Thank you.
uv_unwrap.zip (695 KB)
The one that is open was easy enough to unwrap nicely, but there appear to be some duplicate faces or something that are connecting in a weird way and unwrapping across the uv image editor.
the problem is being caused by the n-gons. I’m not sure why they are doing that, but if you convert the n-gons which are all along the edges to quads, it should unwrap properly.
To quickly convert your ngons into triangles
This way all the ngons (faces made with more than 4 vertices) will be triangulated.
But it may not solve the problem, as the bad repartition of faces will probably challenge the Blender unwrapper and still deform the end result unfortunately.
If Blender had a good remesher, it would be probably fast to fix that, but unfortunately the remesh modifier is very subpar and will not provide the hoped result at all.
I decided to give a try at a manual retopology of your model (look for tutorial on retopology if you never did that, it’s simpler than you think, just a bit time consuming) to get full quads and not some odd face repartition (used the spin tool too)
I uploaded he blend of this, so you can give a look to see if it can help.
http://www.pasteall.org/blend/29502
Not perfect as i didn’t gave too much thought and time, but good enough to show how different a clean mesh will unwrap in comparison to one that has messy topology and faces.
Always try to keep a clean mesh for unwrapping, it will work much better this way
EDIT :
Another version, reworked so you can have the top and the bottom making an actual sphere if you join them (both side on a different layer)
not full quads, but unwrap still well, you can get the blend here :
http://www.pasteall.org/blend/29503
I wish I knew enough to keep a clean mesh. I almost went to C/C++ and raw OpenGL to create this shape. 
The only way I knew to make the shape I did in Blender was using a boolean modifier which intersected two shapes, and then using that as a boolean modifier to a new UV sphere to get the bottom segment. The boolean work introduced many artifacts, most of which I had cleaned up.
I appreciate the advice. I’m working on it. I’ll take a look at your link.
Thank you to all who responded. I was finally able to get a good unwrapping by marking seams at 120 degrees. It actually projected nearly perfectly then. Still a little off.