0

I came across a problem where I find the volume of a pyramid through (length * width * height) / 3. I noticed that there are three numbers being multiplied by each other, then divided by three. This reminded me of Average where you add numbers together then divide by the number of numbers that were added together.

Is there a word used to describe Average with multiplication instead of addition? And if so, what is it?

DMVerfurth
  • 25

1 Answers1

3

There is an average of three numbers $x,y,z$ that involves multiplying them, but $\frac{xyz}{3}$ is not it. An average of three values should have the same units as those values; if $x,y,z$ were measured in meters, then $\frac{xyz}{3}$ would be measured in cubic meters. The correct operation to apply after multiplying three numbers to take an average is to take the cube root. $\sqrt[3]{xyz}$ is an average of three numbers known as the geometric mean.

If you are computing $\frac{\text{length} \cdot \text{width} \cdot \text{height}}{3}$, you are probably not computing the volume of a cube, but of a pyramid with a rectangular base. However, the geometric mean of length, width, and height does have a geometric meaning. If we have a rectangular prism that's $x$ by $y$ by $z$, then its volume is the same as the volume of a cube with side length $s = \sqrt[3]{xyz}$, so in this sense $s$ is the "average side length" of the rectangular prism.

Misha Lavrov
  • 142,276
  • 1
    (+1). Dimensionality / homogeneity reasoning is always really nice. –  Apr 24 '18 at 01:49
  • Oh and sorry, I meant square pyramid not a cube. – DMVerfurth Apr 24 '18 at 02:08
  • Does this work for percentages? Like 100 would be the "average" of 50 and 200 – Rivenfall Feb 04 '23 at 08:36
  • 1
    @Rivenfall Yes, you can say that the geometric average of $50%$ and $200%$ is $100%$. (But you have to be careful, because even when percentages are multiplicative, we can do weird things with them. For example, to find the geometric average of a $50%$ increase and a $200%$ increase, we take $\sqrt{1.5 \cdot 3} \approx 2.12$, which we call approximately a $112%$ increase. "If your stock increases by $50%$ in January and by $200%$ in February, the average increase per month was by $112%$.") – Misha Lavrov Feb 04 '23 at 15:57