Let
be the surface of a cylinder of height
and radius
. (
does not include the flat circular ends of the cylinder.) This Demonstration constructs a set of triangles that tend uniformly to
—yet their total area does not tend to the area of
!
Divide
into
subcylinders (or bands) of height
. Construct
congruent isosceles triangles in each band with vertices at the vertices of a regular
-gon inscribed in the circles at the top and bottom of each band, offset by
.
For any point
in
(except the axis of the cylinder), let
be the axial projection of
onto
. As
, to say that the triangles approximate
uniformly means that for any point
on a triangle and any
(independent of
), there is a
such that for all
,
.
The sum of the areas of the triangles is
.
Depending on how the limit is taken,
can differ. If first
with
held fixed and then
, the limit is
, the expected area of the cylinder. If first
with
held fixed and then
, the limit is infinity. If
and
together so that
is some positive constant
, the limit can be chosen to be any number greater than
.
Therefore
does not have a limit.
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