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.

The surface is known as Schwarz's lantern, Schwarz's polyhedron, or Schwarz's cylinder.