A metallic solid right circular cone is of height 84 cm and the radius of its base is 21 cm.

We have,

Height of the cone, $h=84 \mathrm{~cm}$ and

Base radius of the cone, $r=21 \mathrm{~cm}$

Let the radius of the solid sphere be $R$.

Now,

Volume of the solid sphere $=$ Volume of the solid cone

$\Rightarrow \frac{4}{3} \pi R^{3}=\frac{1}{3} \pi r^{2} h$

$\Rightarrow R^{3}=\frac{r^{2} h}{4}$

$\Rightarrow R^{3}=\frac{21 \times 21 \times 84}{4}$

$\Rightarrow R^{3}=21 \times 21 \times 21$

$\Rightarrow R=21 \mathrm{~cm}$

$\therefore$ Diameter $=2 R=2 \times 21=42 \mathrm{~cm}$

So, the diameter of the solid sphere is 42 cm.