How do you find the length of a diagonal in a cuboid using Pythagoras' Theorem?

To find the length of a diagonal in a cuboid, use the formula: √(l² + w² + h²).

A cuboid is a three-dimensional shape with 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 Pythagoras' 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 d1 of this rectangle can be found using Pythagoras' Theorem: d1 = √(l² + w²). This is because the diagonal forms a right-angled triangle with the length and width.

Next, imagine this diagonal d1 as the base of a right-angled triangle where the height h of the cuboid is the other perpendicular side. To find the length of the space diagonal (d) of the cuboid, you apply Pythagoras' Theorem again, this time in three dimensions. The formula becomes: d = √(d1² + h²). Substituting d1 with √(l² + w²), the formula for the space diagonal is: d = √((l² + w²) + h²), which simplifies to d = √(l² + w² + h²).

So, by using Pythagoras' Theorem twice, you can find the length of the diagonal in a cuboid with the formula √(l² + w² + h²). This method helps you understand how the dimensions of the cuboid relate to the length of its diagonal.