The space diagonal of a cube with an edge length of $4 \, \text{cm}$ is given by $4\sqrt{3} \, \text{cm}$.

To determine the space diagonal of a cube, we can use the formula:

where $d$ represents the space diagonal and $a$ is the edge length of the cube. For our example, the edge length $a$ is $4 \, \text{cm}$. Substituting this value into the formula yields:

Let’s explore the reasoning behind this formula. A cube has three dimensions: length, width, and height, all of which are equal. The space diagonal is the longest diagonal that stretches from one vertex of the cube to the opposite vertex, traversing through the interior of the cube. To derive the space diagonal, we can apply the Pythagorean theorem in three dimensions.

First, we calculate the diagonal of one face of the cube. Since each face is a square with an edge length of $4 \, \text{cm}$, we can use the Pythagorean theorem in two dimensions to find the face diagonal:

Next, we consider this face diagonal as one leg of a right triangle, with the other leg being the edge of the cube, which is also $4 \, \text{cm}$. Using the Pythagorean theorem again in three dimensions, we can determine the space diagonal $d$:

Thus, we confirm that the space diagonal of a cube with an edge length of $4 \, \text{cm}$ is indeed $4\sqrt{3} \, \text{cm}$.