## Steinmetz and Cylinders problem

Hello everyone!

I've been discussing this problem with my uncle for a long time now.

I'm trying to solve an old problem with a dice of size 1 of which you bore 3 holes with a diameter of 1 perpendicular to each other and to the surfaces of the dice. And what is the volume of the remaining dice. I found some answers to this by searching trough the web and made a lot of drawings in SketchUp to visualize it better.
By visualizing I came up with quite a simple solution to this problem but it doesn't give me the right answer and I cant figure what I did wrong.
I came up that by subtracting the bi-cylinder-steinmetz solid from a cylinder I could put 3 of those cylinders around a tri-cylinder-steinmetz(see gif animation). But the math just doesn't seem to work.
Found this equation on wikipedia to calculate the volume of a tri-cylinder-steinmetz: (16 - 8 * sqrt(2.0))r3
And for the bi-cylinder-steinmetz: 16 / 3 * r3
What I expected was that: (Cylindervolume - bicylindersteinmetzvolume) * 3 + tricylindersteinmetzvolume < 1
As the volume of the sorrounding dice is 1.
The numbers I got to was: Cylindervolume = 0.785398 , bicylindersteinmetzvolume = 0.625 , tricylindersteinmetzvolume = 0.585786
Ending up with 1.06698 < 1 ......meh

It looks so good in SketchUp but math just wont work. Were did I go wrong?
(If anyone is interested I can upload .skp file of my playground for visualizing this and a c++ program I made to calculate it.)