How do you calculate the length of a 3D hypotenuse?

To calculate the length of a 3D hypotenuse, use the formula: √(x² + y² + z²).

In more detail, the 3D hypotenuse refers to the diagonal distance between two opposite corners of a rectangular prism (or cuboid). Imagine you have a box, and you want to find the longest straight line that can fit inside it, stretching from one corner to the opposite corner. This line is the 3D hypotenuse.

To find this length, you need the measurements of the box's length (x), width (y), and height (z). These measurements represent the three sides of a right-angled triangle in three dimensions. The formula √(x² + y² + z²) is derived from the Pythagorean theorem, which you might already know for 2D right-angled triangles (a² + b² = c²).

Here's how it works: First, square each of the three dimensions (x, y, and z). Then, add these three squared values together. Finally, take the square root of the sum. This will give you the length of the 3D hypotenuse.

For example, if you have a box with dimensions 3 cm (length), 4 cm (width), and 12 cm (height), you would calculate the hypotenuse as follows:
1. Square each dimension: 3² = 9, 4² = 16, 12² = 144.
2. Add these squared values: 9 + 16 + 144 = 169.
3. Take the square root of the sum: √169 = 13.

So, the length of the 3D hypotenuse is 13 cm.