To determine the length of the diagonal of a 3D rectangle, commonly referred to as a rectangular prism or cuboid, you can utilize the formula:

Where:

• $l$ is the length,
• $w$ is the width,
• $h$ is the height of the cuboid.

A cuboid has three dimensions: length, width, and height. To find the diagonal that stretches from one vertex of the cuboid to the opposite vertex, we can apply the Pythagorean theorem in three dimensions.

To start, consider the diagonal of the base of the cuboid. For a rectangle with length $l$ and width $w$, the diagonal can be calculated using the Pythagorean theorem in two dimensions:

Next, treat this diagonal as the base of a right-angled triangle, with the height $h$ of the cuboid serving as the other perpendicular side. We can now apply the Pythagorean theorem again in three dimensions. The formula for the diagonal of the cuboid becomes:

When we simplify this expression, we arrive at:

This formula enables you to compute the length of the diagonal for any cuboid, provided you know its dimensions: length, width, and height. Simply square each of the dimensions, sum them, and take the square root of the result. This approach is a clear application of the Pythagorean theorem extended into three dimensions.