The volume of a cylinder with a radius of $4$ cm and a height of $10$ cm is approximately $502.65$ cubic centimeters.

To calculate the volume of a cylinder, we use the formula:

In this formula, $V$ represents the volume, $r$ is the radius of the base, and $h$ is the height of the cylinder. For our specific case, the radius $r$ is $4$ cm, and the height $h$ is $10$ cm.

Let’s break down the calculation step-by-step:

1. First, square the radius: $$4^2 = 16$$
2. Next, multiply this result by the height: $$16 \times 10 = 160$$
3. Finally, multiply by $\pi$ (approximately $3.14159$): $$160 \times \pi \approx 502.65$$

Therefore, the volume of the cylinder is approximately $502.65$ cubic centimeters. This means that if the cylinder were filled with water, it would hold about $502.65$ cubic centimeters of water.

Understanding this formula is essential for solving problems related to the volume of cylindrical shapes, which you may encounter in various real-life applications and academic assessments.