I immediately shared it and replied, "You know that I'll have to calculate the value of d which makes that true." But, of course, I messed up and commented on my own post and not his. No matter. I went ahead and figured it out.
A sphere has a Volume of (4/3) pi * r ^ 3. A hemisphere would therefore be (2/3) * pi * r ^ 3.
Factoring out r ^ 2 from this leaves d ^ 2 = (2/3) * pi * r, meaning that d = SQRT( (2/3) * pi * r), which wasn't as bad as I might have expected.
The problem is that d = 2r, or r = (1/2)d. Which means that I had to start over because this is actually a stinkin' quadratic equation. It was not the expression I was looking for.
Anyway, long story shorter, given the limitations of this text medium, here's the answer: r = 0 (discard) or r = pi / 6, which makes d = pi / 3.
The check is left as an exercise to the reader.