# Did you know that there is a 30-60-90 triangle in a hyper cube?

Had a great time this morning with my older son talking about right triangles.  The section we were in today discusses properties of 45-45-90 and 30-60-90 triangles.   He was quite surprised to see the factor of $\sqrt{3}$ arise in the 30-60-90 triangle (and following that in the calculation of the area of an equilateral triangle).

I wanted to expand on that surprise a little and was going to mention that you see $\sqrt{5}$ in a pentagon:

Me:  “Well, you see $\sqrt{2}$ in a square . . . .”

him: “I wonder if you see $\sqrt{3}$ in a cube.”

Oh yes, that’s a much better thing to talk about than seeing $\sqrt{5}$ in a pentagon.    We used a Rubik’s cube and our Zometool set to see if we could get to the bottom of this idea.  We started off the talk with the square as a reminder of where the $\sqrt{2}$ comes from.

Next we took a look at a cube to see if we could find any $\sqrt{3}$‘s hiding anywhere:

Now, just for fun, what about a hyper cube?  I wish I had better drawing skills, but at least this shows the idea.  I did not know that the long diagonal of a hyper cube forms a 30-60-90 triangle, so that was a neat surprise connection with what we were taking about today:

So, a fun and unexpected little project about square roots.  Probably would be neat to revisit something like this whenever we start talking about the distance formula since this project essentially shows how you calculate distances in dimensions higher than 3.  Nice morning.