### By Mark Dawes (December 2020)

What is your response to the instruction “draw the net of a cylinder”?

To draw a rectangle and two circles

To quibble with the word “the” in the question

To state that “the net of a cylinder” doesn’t exist

Let’s look at each of these in turn.

a) Here, surely is a net of a cylinder:

It features the three shapes we need, in the correct configuration, so it must be a net.

b) But we can’t talk about the net of a shape, because there are lots of different possibilities. If we move one (or both) of the circles up or down the sides of the rectangle then it will still behave in the same way.

There are two nets of a tetrahedron (triangle-based pyramid):

There are 11 nets of a cube, 11 nets for an octahedron and, amazingly, 43380 different nets for a dodecahedron!

So we should instead refer to ‘a net’ and not ‘the net’ of a shape.

But what about answer (c)? Sure the net exists? There are some potential problems with the idea of having a net for a cylinder.

It is uncontroversial that the blue diagram is a net of a cube.

But what about the green one? It isn’t one of the 11 nets and I think most people would discount it because the bottom-right square isn’t properly connected to the rest of the shape.

Apart from anything else, we don’t know which way that square will go.

If it rotates one way and the two yellow edges are joined then it will make a cube, but if it goes the other way and the two orange edges join then it won’t.

If this is discounted because the two right-most squares only join at a point, then the cylinder should be discounted as having a net too, because the sides of the rectangle are tangents to the circles, so they touch only at a single point.

This means that a common definition of a net: “The 2D faces that are put together to make a 3D shape” is not correct.

In fact, even this, more common definition is dubious: “A 2D shape you can fold up to make a 3D shape”, because this next diagram does hold together and can be folded to make a shape (what would you call it?) but isn’t actually a net:

This comes back to the question of ‘who defines maths’? And how do they define a net?

For definitions, I think MathWorld is authoritative (though the level is above that of school maths). Their definition involves the idea that a net refers only to something that can make a polyhedron (ie a 3D shape with flat faces). Their definition is not appropriate for GCSE maths but the Wikipedia entry for net(polyhedron) is more useful:

In geometry a net of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron.

With some minor tweaking, this could be used with students.

The crucial thing is that in order to have a net, the shape must be a polyhedron. A cylinder, therefore, doesn’t have a net (and neither does a cone).

The important bit!

What was your reaction to finding out that there is no such thing as the net of a cylinder?

Denial? Don’t be silly – of course the thing with circles either side of a rectangle is a net of a cylinder.

Rejection? Pfft – what does it matter? It still works like a net, so even if it technically isn’t then that’s not really a problem.

Annoyance? Why didn’t anybody teach me this properly?

Bewilderment? How was I supposed to know that: there isn’t a recognised set of maths definitions?

Anger? But that doesn’t make sense! We need a net for a cylinder!

Frustration? Look – when I Google ‘net of a cylinder’ I get a quarter of a million hits!

How do our students feel when they discover something they thought they knew, actually turns out not to be accurate? What can we do to help them in such circumstances?

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