"rad" is verbal short hand for "the square root of". Sorry for the confusion.

I was wrong in both of my prior answers. I misremembered how a 30-60-90 triangle works. I've done some searching and think I have the solution now:

1) Draw an equilateral triangle within a circle.
2) from the center of a circle, draw a line to any vertex of the triangle.
3) from the center of a circle, draw a line to the midpoint of one side of the triangle. It should meet at a 90 degree angle. It should create a 30-60-90 triangle. The middle sized side of the triangle (not the shortest nor the longest) will be one half the size of the big equilateral triangle side-length.
4) With Pythagoras in the background: the ratio of sizes of the sides of a 30-60-90 triangle are 1:1/2:(rad-3)/2. The side we are interested in (side that makes up half of the equilateral triangle's side) is the rad 3 side. Since the radius of our circle is on the "1" side, the size of the side we are interested in is r(rad-3)/2. Since we need two of these to make up the side of the equilateral triangle, the size of that side is r(rad-3).
5) With Pythagoras in the foreground: going back to that same 30-60-90 triangle, the equation r*2 = (r/2)*2 + n*2. (n is the side we are trying to calculate the size of). Solving for n gives us r(rad-3)/2 (again). Multiply by 2 to get the size of the equilateral triangle and you get r(rad-3).