Calculator · Engineering Mechanics
Huygens–Steiner (Parallel-Axis) Theorem Calculator
Also known as the parallel-axis theorem, the Huygens–Steiner theorem moves a centroidal moment of inertia to a parallel axis. Its standard form starts from the centroidal or centre-of-mass axis, and its transfer term is non-negative.
Quantity
Inputs
I_c is used as both the centroidal Ix and Iy so each transfer is shown independently; a full section with distinct Ix, Iy and Ixy belongs in the Area Moment of Inertia tool.
Results
| Transfer term A·dy² | 20000 cm⁴ |
|---|---|
| Ix = I_c + A·dy² | 21670 cm⁴ |
| Transfer term A·dx² | 0 m⁴ |
| Iy = I_c + A·dx² | 1667 cm⁴ |
| Ixy = Ixy,c + A·dx·dy | 0 m⁴ |
Technical reference
Ix = I_c + A·dy² (area, and likewise Iy = I_c + A·dx²)
I = I_G + m·d² (mass)
- I_c, I_G
- second moment of area or mass moment of inertia about the CENTROIDAL / centre-of-mass axis[m⁴ or kg·m²]
- A, m
- area or mass of the body[m² or kg]
- d (dx, dy)
- perpendicular distance between the two parallel axes[m]
The standard form starts from an axis through the centroid (area form) or centre of mass (mass form) and transfers to a parallel axis at perpendicular distance d; it does not directly transfer between two arbitrary axes.
Also known as the Huygens–Steiner theorem, hence the page title; because A·d² or m·d² is non-negative, the centroidal or centre-of-mass axis has the minimum inertia within its family of parallel axes.
Assumptions
- · The starting value must be about the CENTROIDAL or centre-of-mass axis; direct transfer between two non-centroidal axes is outside this form.
- · The two axes are parallel, and d is the perpendicular distance between them.
- · Area [m⁴] and mass [kg·m²] forms are kept separate and are not interchangeable.