A PSA on Cylinders and Prisms

OK, listen up, there seems to be some confusion about this. I shall make the point several times here, but first I shall make it in bold:

Cylinders are not prisms.

In fact, it’s the other way around.


A cylinder is a 3D shape that has parallel end-faces and a cross-section that’s “the same” all the way down. Corresponding points on the two end faces are joined by parallel lines.

A cylinder does not need to be circular. The sides do not need to be vertical – a parallelepiped1, for example, is (strictly speaking) a cylinder: it follows all the rules a cylinder has to follow.

(As an aside: I consider a square to be a rectangle. A rectangle is a parallelogram whose sides are at right angles; a square is a rectangle with equal-length sides. The set of squares is a subset of the set of rectangles. I’m sure we could come up with an Euler diagram.)

What comes to mind when one thinks of a cylinder - most likely - is also a cylinder, you’ll be pleased to know. In particular, it’s a right circular cylinder. Right means the sides are vertical, at right-angles to the base (our parallelepiped is not a right cylinder) and circular means… well, have a guess.

So where does that leave prisms?

Prisms are also a subset of cylinders - precisely, the ones that have polygons for cross-sections. Prisms are polyhedra by definition: a circular cylinder is not a prism.

Our parallelepiped is a prism - it’s a polyhedron, the cross-section is the same shape everywhere, and if I take any two points on the perimeter of the top face, and join them to their corresponding points on the bottom, I get two parallel lines.

It’s not a right prism, of course - unless you were to ensure that one of the faces was directly above another.

But the FORMULA SHEET!

The formula sheet is wrong.

Or rather, it’s wrong by implication - the shape they have shown is not a prism, it’s a (right) cylinder. It’s a bit needless: the formula for the volume of a prism is just the same as the one for the volume of a cylinder. (In fact, it doesn’t even need to be a right prism/cylinder, as long as you use the vertical height rather than the slant height.)

Does it really matter?

OF COURSE IT MATTERS! PEOPLE ARE USING WORDS WRONG!

Definitions are critical in maths. It is possible to mess around with them, if it’s convenient and clear and you know what you’re doing. However, I’ve never seen any sort of justification for redefining prisms to include non-prisms, and can’t imagine why one would want to.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

  1. a shape with pairs of parallel parallelograms for faces []

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