What is the method for calculating the diagonal of a 3D rectangle?

To calculate the diagonal of a 3D rectangle, use the formula: \( \sqrt{l^2 + w^2 + h^2} \).

In more detail, a 3D rectangle, also known as a rectangular prism or cuboid, has three dimensions: length (l), width (w), and height (h). To find the length of the diagonal that stretches from one corner of the cuboid to the opposite corner, you can use the Pythagorean theorem in three dimensions.

First, consider the diagonal on the base of the cuboid. If you have a rectangle with length \( l \) and width \( w \), the diagonal of this rectangle can be found using the Pythagorean theorem in two dimensions: \( \sqrt{l^2 + w^2} \).

Next, treat this diagonal as the base of a right-angled triangle where the height \( h \) of the cuboid is the other perpendicular side. Now, apply the Pythagorean theorem again in three dimensions. The formula becomes:
\[ \text{Diagonal} = \sqrt{(\sqrt{l^2 + w^2})^2 + h^2} \]

Simplifying this, you get:
\[ \text{Diagonal} = \sqrt{l^2 + w^2 + h^2} \]

This formula allows you to calculate the length of the diagonal of any cuboid if you know its length, width, and height. Just square each of the dimensions, add them together, and then take the square root of the result. This method is a straightforward application of the Pythagorean theorem extended into three dimensions.