The volume of a sphere with a radius of $5 \, \text{cm}$ is approximately $523.6 \, \text{cm}^3$.

To calculate the volume of a sphere, you can use the formula:

In this formula, $V$ represents the volume, and $r$ denotes the radius of the sphere. For our case, the radius $r$ is $5 \, \text{cm}$. Substituting the radius into the formula, we have:

First, we need to compute $5^3$, which is calculated as follows:

Next, we multiply this result by $\pi$ (approximately $3.14159$):

Now, we perform the multiplication of $125$ by $\pi$:

Finally, we multiply this result by $\frac{4}{3}$:

Thus, the volume of the sphere is approximately $523.6 \, \text{cm}^3$. This formula is fundamental in geometry and illustrates the relationship between the radius and the volume of a sphere. It is important to note that the volume increases significantly as the radius increases, due to the cubic relationship in the formula. Therefore, even a slight increase in the radius leads to a much larger volume.