What is the formula for calculating the diagonal of a rectangular prism?

The formula for calculating the diagonal of a rectangular prism is: \( \sqrt{l^2 + w^2 + h^2} \).

To understand this formula, let's break it down. A rectangular prism, also known as a cuboid, has three dimensions: length (l), width (w), and height (h). The diagonal we are interested in is the longest line that can be drawn from one corner of the prism to the opposite corner, passing through the interior of the prism.

First, consider the base of the prism, which is a rectangle. The diagonal of this rectangle can be found using Pythagoras' theorem. If you have a rectangle with length \( l \) and width \( w \), the diagonal \( d_1 \) of the base is given by:
\[ d_1 = \sqrt{l^2 + w^2} \]

Next, imagine this diagonal \( d_1 \) as the base of a right-angled triangle, where the height \( h \) of the prism is the other perpendicular side. To find the diagonal \( D \) of the entire prism, we apply Pythagoras' theorem again:
\[ D = \sqrt{d_1^2 + h^2} \]

Substituting \( d_1 \) into this equation, we get:
\[ D = \sqrt{(\sqrt{l^2 + w^2})^2 + h^2} \]
\[ D = \sqrt{l^2 + w^2 + h^2} \]

So, the formula \( \sqrt{l^2 + w^2 + h^2} \) allows you to calculate the diagonal of a rectangular prism by considering the contributions of all three dimensions. This formula is a direct application of Pythagoras' theorem in three dimensions, making it a powerful tool for solving problems involving 3D shapes.