To determine the volume of a sphere with a diameter of $8 \, \text{cm}$, we can utilize the formula for the volume of a sphere, which is given by:

First, we need to calculate the radius of the sphere. The radius is half of the diameter. Given that the diameter is $8 \, \text{cm}$, the radius $r$ can be calculated as follows:

Next, we substitute the radius into the volume formula:

Substituting $r = 4 \, \text{cm}$ into the equation gives:

We need to calculate $4^3$, which represents $4$ cubed (or $4$ multiplied by itself three times):

Now we can update our volume formula:

Next, we multiply $\frac{4}{3}$ by $64$:

Thus, the volume of the sphere can be expressed as:

To obtain a numerical approximation, we can use $\pi \approx 3.14$:

Therefore, the volume of the sphere is approximately $268.08 \, \text{cm}^3$.