The elementary geometric lemma tells us how small to make K in order to make sure that no two snipped-away parts overlap. By choosing
we can guarantee that any relevant hemisphere with which C
conflicts is within a distance 
of C.
(See Figure 18.)
Figure 18: Making sure conflicting hemispheres are close.
Thus by the elementary geometric lemma if two hemispheres both
conflict with C the distance between the centers of the corresponding
snips must be 
.
(Recall that we only have to worry about relevant hemispheres,
which have height 
.)
But by choosing
we can guarantee that the distance from center to tip of either snip
is 
.
(See Figure 19.)
Figure 19: Making sure snips are short.
Thus if the snips overlap, the distance between their centers must be
,
a contradiction.
For the values
and
,
we find that these conditions on K
will be satisfied for
.
With this value of K,
each hemisphere is asking for growth room of
th the area that its tubes cover at height h.