# 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.

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).

### 4 responses to “Questions from middle-schoolers VIII: Prisms”

1. Alex Otto

My students say thank you for the straight answer :) They say they would like to request a baked “pi” from you.

2. Alex Otto

I want you to know there was a small discussion about what to leave for a comment (We responded during our math class.) Several students thought we should the comment to ask another mathematical question. However, since we met before lunch today (usually we meet after lunch), the request for a baked “pi” won out over the request for more mathematical knowledge. (Gotta love middle schoolers! :)

3. Alex Otto

Oops – that should be “Several students thought we should USE the comment. . “

4. I have often found it useful to explain prisms as “extrusions”, similar to what most kids did with Play-Dough when they were young. By thinking of them this way, it is easy to figure out how to name them: what was the shape through which they were extruded?

It is also easy to figure out the formula for their volume: surface area of the extrusion shape times the distance it was extruded:
Circle => circular prism => cylinder => (pi)(radius^2)(distance)
Square => square prism => (side^2)(distance)
Rectangle=> rectangular prism => (length)(width)(distance)
Triangle => triangular prism => (1/2)(base)(altitude)(distance)
Trapezoid => trapezoidal prism => (1/2)(base1 + base 2)(altitude)(distance)