The volume of a solid hemisphere is numerically equal to its total surface area. Its radius is:

Let the radius of the solid hemisphere be $r$.
The volume of a solid hemisphere is given by:

$$V=\frac{2}{3}\pi r^3$$

The total surface area (TSA) of a solid hemisphere is given by:

$$\text{TSA}=3\pi r^2$$

According to the question, the volume is numerically equal to the total surface area:

$$\frac{2}{3}\pi r^3=3\pi r^2$$

Dividing both sides by $\pi r^2$ (since $r\ne 0$):

$$\frac{2}{3}r=3$$

$$r=3\times \frac{3}{2}=\frac{9}{2}=4.5=4\frac{1}{2}\text{ units}$$