Box Packing Calculator
Compare all six orientations for identical rectangular items in a rectangular container. Find the best uniform grid, counts along each axis and volume utilization.
YOUR CALCULATION
Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
For each of the six permutations, divide each container axis plus one gap by the corresponding item dimension plus one gap, then round down. Multiply the three axis counts and compare orientations.
n_axis=floor((container_axis+gap)/(item_axis+gap)); count=nₓnᵧn_z; utilization=count×item_volume/container_volume
Worked example
A 60×40×30 cm container fits 24 items measuring 20×10×15 cm at zero clearance: 3×4×2. All six orientations are compared, with ties shown in the table.
Assumptions and input details
This compares uniform axis-aligned grids with every item in the same orientation; it does not claim a globally optimal mixed-orientation packing. Clearance applies only between neighbors, not at container walls. Use usable internal container dimensions and external item dimensions; no wall thickness, padding, weight limit, loading access or deformation is inferred. A zero count means no item fits in that orientation. The model supports at most 1e12 positions per axis and items per orientation, dimensions up to 1e6 m, and floating-point rounding near exact fits. Volume utilization excludes gaps from item volume.
Unit definitions
Unit selection is independent of language. Each selector contains only units for its physical quantity. Calculations use defined base units internally.
Choose from 23 length definitions, including explicitly labelled regional measures. Area conversions square the length factor; geometric volumes cube it.
Regional measures are not interchangeable merely because their names look similar. This is a documented selection, not a claim to include every historical unit.
NIST measurement tables · Japanese traditional units · Mainland China and Hong Kong unit distinctions
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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