To calculate the length of a diagonal in a cuboid, you can use the formula:

where $l$ represents the length, $w$ the width, and $h$ the height of the cuboid. A cuboid is a three-dimensional shape defined by these three dimensions.

To determine the diagonal that extends from one corner of the cuboid to the opposite corner, we can apply Pythagoras’ Theorem in three dimensions.

First, consider the diagonal of the base of the cuboid. If you look at the rectangular base with length $l$ and width $w$, the diagonal $d_1$ of this rectangle can be calculated using Pythagoras’ Theorem:

This calculation is valid because the diagonal forms a right-angled triangle with the length and width of the rectangle.

Next, envision this diagonal $d_1$ as the base of another right-angled triangle, where the height $h$ of the cuboid serves as the other perpendicular side. To find the length of the space diagonal $d$ of the cuboid, we apply Pythagoras’ Theorem once more, this time in three dimensions. The formula is:

By substituting $d_1$ with $\sqrt{l^2 + w^2}$, we arrive at:

This simplifies to:

In summary, by applying Pythagoras’ Theorem twice, you can find the length of the diagonal in a cuboid using the formula:

This approach illustrates the relationship between the cuboid’s dimensions and the length of its diagonal.