Tag Archives: prisms

Questions from middle-schoolers VIII: Prisms

Is it still a triangular prism when it sits on a rectangular side?

A prism is a 3d figure and each is named after its base. Below is a triangular prism.

If you rotate it so that the rectangular face is on the bottom, as below, why is it still called a triangular prism if technically its base is no longer a triangle?

This is a really important question. In geometry it is essential to think of figures by their geometric properties. Where these figures are and what direction they are facing are not geometric properties.

In the everyday world, we often get these confused. Think of a baseball diamond. We call it a “diamond” because it sits on its corner.

Two squares-one on edge, one on a corner

These are both squares.

But in geometry that doesn’t matter. A baseball “diamond” is really a square; it has four sides all the same length and it has right angles. That’s what it takes to be a square in geometry-it doesn’t matter what direction the sides are running, nor where the square is.

So your triangular prism is a triangular prism no matter what direction it is pointing. And the triangular bases are still called the “bases” of the prism even when the prism isn’t sitting on them.

A better way to talk about a triangular prism would be that it can be placed on a triangular base-not that is has to be resting on a triangular base already.

By the way, the rectangle-as-base orientation of a triangular prism suggests another formula for volume. If we put two of these prisms together (as below), we get a solid whose volume is (Area of rectangular base) x (height of prism).

So the volume of a triangular prism is half of that: (1/2) x (Area of rectangular base) x (height of prism).