It turns out that to insure that snipped-away parts don't overlap,
it is sufficient to make sure that the annuli
that the hemispheres try to annex are sufficiently small,
though of course we must still make them large enough to be useful.
Specifically,
say that to the hemisphere 
,
with notation as above,
we associate an annulus of outer radius
where K is some universal constant yet to be specified. The claim is that if K is chosen small enough, then no two snipped-away parts will ever overlap. The truth of this statement depends on only the grossest of estimates of how densely one can pack circles in the plane.