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Calculate the space diagonal of rectangular boxes using length, width, and height
Space Diagonal: d = √(length² + width² + height²)
The space diagonal (or body diagonal) of a rectangular box is the longest straight line that can be drawn inside the box, connecting two opposite vertices through the interior. It extends through 3D space, unlike face diagonals which lie on a surface.
Use the 3D Pythagorean theorem: d = √(l² + w² + h²), where l is length, w is width, and h is height. For example, a box that's 3×4×12 has diagonal √(9+16+144) = √169 = 13 units.
A face diagonal lies on one of the box's faces (2D surface) and connects two opposite corners of that face. A space diagonal goes through the interior of the box (3D) and connects two opposite vertices of the entire box. Space diagonal is always longer.
The 2D Pythagorean theorem (a² + b² = c²) extends to 3D as a² + b² + c² = d². First, find a face diagonal using 2D Pythagorean, then use that diagonal with the third dimension to find the space diagonal.
Shipping companies use diagonal measurements to determine if a package fits in cargo spaces or through doorways. The diagonal represents the minimum clearance needed to maneuver a box through an opening at any angle.
No, volume alone is insufficient. Infinite box dimension combinations can produce the same volume but different diagonals. You need at least two dimensions plus volume, or all three dimensions.
For a cube with side length s, the space diagonal is s√3. This comes from d = √(s² + s² + s²) = √(3s²) = s√3. For a 1-unit cube, the diagonal is approximately 1.732 units.
The formula d = √(l² + w² + h²) only works for rectangular boxes (right angles). For irregular shapes, you'd need to use vector mathematics or break the shape into rectangular sections and calculate accordingly.