Looking up "volume of spherical cap" in many places (for example, here: http://mathworld.wolfram.com/SphericalCap.html) if the radius is $r$ and the height from the end of the sphere is $h$, then the volume is $\frac13\pi h^2(3r-h)$.

Since the volume of the sphere is $\frac43 \pi r^3$, you want the volume of the cap tp be one-third of this, or $\frac13\pi h^2(3r-h) =\frac49 \pi r^3 $ or $3 h^2(3r-h) =4 r^3 $. Dividing by $r^3$, this becomes $3(\frac{h}{r})^2(3-\frac{h}{r}) = 4 $. Finally, letting $x = \frac{h}{r}$, this is $3x^2(3-x) = 4$ or $3x^3-9x^2+4 = 0 $.

According to Wolfram Alpha, the root of this is $x\sim 0.77393$, so $d = 2(1-h) \sim 2-1.54786 = .45214 $, which agrees very nicely with your computation.

Looking up "volume of spherical cap" in many places (for example, here: http://mathworld.wolfram.com/SphericalCap.html) if the radius is $r$ and the height from the end of the sphere is $h$, then the volume is $\frac13\pi h^2(3r-h)$.

Since the volume of the sphere is $\frac43 \pi r^3$, you want the volume of the cap to be one-third of this, or $\frac13\pi h^2(3r-h) =\frac49 \pi r^3 $ or $3 h^2(3r-h) =4 r^3 $. Dividing by $r^3$, this becomes $3(\frac{h}{r})^2(3-\frac{h}{r}) = 4 $. Finally, letting $x = \frac{h}{r}$, this is $3x^2(3-x) = 4$ or $3x^3-9x^2+4 = 0 $.

According to Wolfram Alpha, the root of this is $x\sim 0.77393$, so $d = 2(1-h) \sim 2-1.54786 = .45214 $, which agrees very nicely with your computation.