I am sitting in my office worn down a bit from just coming in from mowing my yard in the Texas 90-degree heat and humidity…

My thoughts drifted to my three-D printers I have in my office. They need to be making something.

I have a need for a small box to store the 25 MM diameter key rings I use when making key fobs. I just ordered in another 100. They come in a vinyl-like pouch but I would like to store them in a printed box I usually make. However, the volume (capacity) of the box is a bit too small for 100 rings.

A real easy solution with 3D printing is the 3D graphic file used to print something can very easily be scaled up or down dimensionally by percentage in the printing slicer software.

So I start thinking… (very dangerous) … what do I need to scale up my little box to hold twice as much volume? Hum… If I make it twice as high, that’s the easy answer. Right? (Yes)

But I don’t want a box that is twice as high. I just want to expand all three directions equally enough to make the volume of the box double.

In my brain, I am intuitively thinking, if I multiply each dimension 125% or make then 25% longer, that should do it. It’s close but not the perfectly right answer. I was trying to avoid figuring cube roots in my head. Easy? Yeah, you try it…

Here’s the math thought problem…

The goal is to double the volume of the existing box. We don’t need to know the actual volume or even the actual dimensions (inside or out) of that box. We have a drawing with no printed dimensions but the ability to scale all three existing dimensions with a computer program by plugging in the correct percentage increase value.

What I want is to go from 100% - what I have, to 200% (double). Volume is a cubic value. That is the clue. What equal percentage of dimensional increase on width, length and height will give me twice or two (2) times the volume? Hint - I need the *cube* root of 2.

Looking it up (Hey this is the computer age!) the cube root of 2 is 1.25992104989.

My intuitive brain picture was off by 0.00992+ or almost 1% so that’s not too bad. I used 125% increase for each dimension and that will do the job I want.

Three times the volume requires the cube root of 3, which is 1.44224957031, pretty far from what I might intuitively guess. Cube roots can get a bit weird. Here is a Wikipedia Link.

Doubling each ** dimension** produces eight (8) time the volume. 2x2x2=8 Cube root of 8 is 2. That makes it easy for my brain to realize how the math works. Everything from 1 to 8 times the volume is going to be more than one hundred percent and less than two hundred percent multiplier for all three dimensions.

Our language of percentage can be confusing. In two dimensions (plane geometry), surface area increases as the square of the distance. In 3 dimensions (solid gepmetry) volume increases as the cube of the distance.

The brown box is the original size. The black box has twice the volume of the brown.