What is the formula for finding the hypotenuse in a 3D right-angled triangle?

The formula for finding the hypotenuse in a 3D right-angled triangle is the square root of the sum of the squares of its three sides.

In a 3D right-angled triangle, also known as a right-angled tetrahedron, you have three sides that meet at a right angle. These sides can be thought of as the edges of a box or a cuboid. To find the hypotenuse, which is the longest diagonal running through the box, you need to use a formula that extends Pythagoras' Theorem into three dimensions.

The formula is: \( d = \sqrt{x^2 + y^2 + z^2} \), where \( d \) is the length of the hypotenuse, and \( x \), \( y \), and \( z \) are the lengths of the three sides that meet at the right angle.

To understand this better, imagine you have a cuboid with sides of lengths \( x \), \( y \), and \( z \). First, you find the diagonal on the base of the cuboid using Pythagoras' Theorem in 2D: \( \sqrt{x^2 + y^2} \). This gives you the length of the diagonal on the base. Then, you treat this diagonal as one side of a right-angled triangle, with the third side being the height \( z \). Applying Pythagoras' Theorem again, you get \( d = \sqrt{(\sqrt{x^2 + y^2})^2 + z^2} \), which simplifies to \( d = \sqrt{x^2 + y^2 + z^2} \).

This formula is very useful in various fields, including physics, engineering, and computer graphics, where you often need to calculate distances in three-dimensional space.