Find the Space Diagonal Formula: Expressing Length in Terms of x, y, and z

To solve for the diagonal of a rectangular prism with dimensions $x$, $y$, and $z$, we'll utilize the Pythagorean theorem in three dimensions. This enables us to account for the three different sides of the prism.

Let us break this down into steps:

• Step 1: Identify the dimensions given: length $x$, width $y$, and height $z$.
• Step 2: Recognize that the diagonal stretches across the prism from one corner to the opposite corner, forming a 3D hypotenuse.
• Step 3: Apply the formula for the diagonal of a rectangular prism:

This formula arises because the diagonal spans across the 3D space of the prism. By applying the Pythagorean theorem first to the base rectangle and then incorporating the height, we account for all dimensions of the prism.

Thus, the length of the diagonal is given by $\sqrt{x^2 + y^2 + z^2}$.