# 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 parallelepiped ((a shape with pairs of parallel parallelograms for faces)), 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.