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Calculate box dimensions, volume, surface area, and diagonal. Find a missing dimension from volume.
If you know volume and two dimensions: Missing = Volume ÷ (dim₁ × dim₂). For example, if V = 100 and length = 5, width = 4, then height = 100 ÷ (5 × 4) = 5.
The space diagonal: d = √(l² + w² + h²). This is the distance between opposite corners, the longest straight line inside the box.
The side length of a cube with the same volume. If your box has volume V, the equivalent cube has side = ∛V. A cube has the minimum surface area for a given volume.
Shipping companies use dimensional weight = (L × W × H) ÷ divisor. Common divisors: 139 for inches (domestic) or 166 (international), 5000 for cm (metric).
For a given volume, a cube uses the least cardboard (surface area). A 10×10×10 box uses less material than a 5×10×20 box with the same 1000 cubic unit volume.
Surface area = 2(lw + lh + wh) plus extra for flaps, overlaps, and seams. Typical box patterns require 10-20% more material than the surface area.