To calculate the length of the diagonal in a three-dimensional rectangle, commonly referred to as a cuboid, we utilize the 3D Pythagorean theorem. This theorem is an extension of the 2D Pythagorean theorem, which applies to flat surfaces.

For a cuboid with side lengths denoted as $a$, $b$, and $c$, the formula for the diagonal $d$ is given by:

$d = \sqrt{a^2 + b^2 + c^2}$

Consider a cuboid with the following dimensions: $5 \, \text{cm}$, $12 \, \text{cm}$, and $13 \, \text{cm}$. We can assign these values to the variables as follows:

$a = 5 \, \text{cm}$
$b = 12 \, \text{cm}$
$c = 13 \, \text{cm}$

Next, we will substitute these values into the diagonal formula:

$d = \sqrt{5^2 + 12^2 + 13^2}$

Now, we will calculate the squares of each side:

$5^2 = 25$
$12^2 = 144$
$13^2 = 169$

We then sum these squared values:

$25 + 144 + 169 = 338$

Finally, we compute the square root of the total:

$d = \sqrt{338}$

Using a calculator, we find:

$d \approx 18.39 \, \text{cm}$

Thus, the length of the diagonal in the cuboid is approximately $18.39 \, \text{cm}$. This method effectively accounts for all three dimensions of the cuboid, ensuring an accurate calculation of the diagonal length.