Parallel Axis Theorem Calculator
Shift an area second moment from a centroidal axis to a parallel axis using independently selected property, area, offset and output units.
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
Convert the centroidal area second moment, section area and perpendicular offset to SI powers. Add the nonnegative transfer term A d², then convert the fourth-power result to the selected output base length.
I_parallel=I_centroid+A d²
Worked example
For a 100 mm × 100 mm rectangle, I_centroid=8,333,333.333 mm⁴ and A=10,000 mm². Shifting 50 mm gives I=33,333,333.333 mm⁴.
Assumptions and input details
This page shifts an area second moment (geometric second moment), not a mass moment of inertia. The supplied centroidal axis and target axis must be parallel, and the offset is their perpendicular distance. Area must be positive; centroidal property and distance are nonnegative. Product of inertia, rotated axes, composite centroid finding, section modulus, stress, stiffness and sign conventions for holes are outside this one-section transfer.
References: MIT: parallel axis theorem for section moments
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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