Moment of inertia of a rectangle
The textbook formula b·h³/12 for the centroidal axis, plus b·h³/3 about the base via the parallel-axis theorem. Square and hollow-box variants. Worked examples for the 100 × 200 mm reference case. Reviewed by a licensed PE.
Use the rectangle moment of inertia calculator
The calculator opens with the rectangle form selected. Enter width b and height h to get Ix, Iy, polar J, section modulus Sx, plastic Zx, radius of gyration, area, and centroid — all in one panel. Tick the "Parallel-axis" checkbox to compute about an offset axis (e.g. the base or any other line parallel to the centroidal axis).
Pick an AISC W / IPE / HEA / HEB / HSS / channel / angle to auto-fill dimensions, or leave on "Manual entry" for a custom section.
Rectangle: simplest case. Ix = b·h³/12 about centroidal axis parallel to base.
Hollow rectangle (centred). Ix = (b·h³ − bi·hi³)/12. Ensure inner dims < outer.
Solid circle. Ix = Iy = π·D⁴/64. Polar J = π·D⁴/32.
Hollow circle / pipe. Ix = π(D⁴ − d⁴)/64. d = inside diameter.
Half-disc with flat edge at bottom. Centroid sits 4r/(3π) above the flat edge.
I-beam (W / IPE / UB). Symmetric flanges and centred web.
T-beam: top flange + vertical stem. Centroid is below the flange — origin is bottom of stem.
C-channel: web on left, flanges open to the right. Asymmetric — centroid offset from web.
L-angle: legs of width leg1 (horizontal) and leg2 (vertical), uniform thickness t.
Right triangle: right-angle at the origin, base b along x, height h along y. Centroid at (b/3, h/3).
Isosceles trapezoid: a = top, b = bottom, h = height. ȳ = h(b + 2a)/(3(a+b)) from bottom.
- Polar J
- — mm⁴
- Sx (elastic)
- — mm³
- Sy (elastic)
- — mm³
- Zx (plastic)
- — mm³
- rx
- — mm
- ry
- — mm
- Area
- — mm²
- Centroid (x̄, ȳ)
- —
- I about offset
- —
The rectangle moment of inertia formula
Three forms cover every common reference axis. The first is the centroidal one — the canonical b·h³/12 — and the other two use the parallel-axis theorem for non-centroidal references.
- I_x
- about horizontal centroidal axis (parallel to base b), mm⁴ / in⁴
- I_y
- about vertical centroidal axis (parallel to side h), mm⁴ / in⁴
- b
- width (horizontal), mm / in
- h
- height (vertical), mm / in
- I_base
- I about an axis at the bottom edge, parallel to b, mm⁴ / in⁴
- A
- area = b·h, mm² / in²
- d
- distance from centroidal axis to base = h/2, mm / in
- b, h
- outer dimensions, mm / in
- b_i, h_i
- inner cavity dimensions (centred), mm / in
Worked example, 100 × 200 mm rectangle
Compute every common property of a 100 mm wide × 200 mm tall rectangle. Mental-check numbers — useful for sanity-checking the calculator.
| Property | Formula | Substitution | Result |
|---|---|---|---|
| Area A | b · h | 100 × 200 | 20 000 mm² |
| Ix (centroidal) | b·h³/12 | 100 × 200³ / 12 | 6.67 × 10⁷ mm⁴ |
| Iy (centroidal) | h·b³/12 | 200 × 100³ / 12 | 1.67 × 10⁷ mm⁴ |
| I about base | b·h³/3 | 100 × 200³ / 3 | 2.67 × 10⁸ mm⁴ |
| Sx (elastic) | Ix / (h/2) | 6.67e7 / 100 | 6.67 × 10⁵ mm³ |
| Zx (plastic) | b·h²/4 | 100 × 200² / 4 | 1.00 × 10⁶ mm³ |
| rx | h/√12 | 200/√12 | 57.7 mm |
| ry | b/√12 | 100/√12 | 28.9 mm |
Note: Ix = 4 × Iy. Doubling the height in the bending direction more than doubles the stiffness — a 4× difference for only a 2× geometric ratio. The shape factor Zx/Sx = 1.5 for any rectangle (often quoted in plastic-design textbooks).
Variants and special cases
Square cross-section
A square is just a rectangle with b = h = a. Substitute into the rectangle formula and both Ix and Iy reduce to the same value: Ix = Iy = a⁴/12. Polar moment J = a⁴/6. Section modulus S = a³/6. The square's symmetry means it has no "strong" or "weak" axis — orientation does not change stiffness. Used for axles, columns where bending direction is unknown, and decorative sections. For a 100 mm square: Ix = 100⁴/12 = 8.33 × 10⁶ mm⁴.
Comparing a square and a 1:2 rectangle of the same area (a = 141 mm square vs 100 × 200 rect, both 20 000 mm²): the rectangle has Ix = 6.67 × 10⁷ mm⁴ while the square has Ix = 3.33 × 10⁷ mm⁴ — the rectangle is 100% stiffer in its strong direction. This is why beams are never square.
Related concepts on this site
Frequently asked questions
- What is the moment of inertia of a rectangle?
- For a solid rectangle of width b and height h, the moment of inertia about the centroidal axis parallel to the base is Ix = b·h³/12. About the perpendicular centroidal axis it is Iy = h·b³/12. For a 100 mm × 200 mm rectangle: Ix = 100 × 200³ / 12 = 6.67 × 10⁷ mm⁴, Iy = 200 × 100³ / 12 = 1.67 × 10⁷ mm⁴. The taller axis dominates by a factor of four — orient the rectangle with its long side vertical to maximise bending stiffness against gravity loads.
- What is the formula for moment of inertia about the base of a rectangle?
- Use the parallel-axis theorem: I about the base = Icentroidal + A · d², where d = h/2 is the distance from the centroid to the base. Plugging in: Ibase = b·h³/12 + (b·h)·(h/2)² = b·h³/12 + b·h³/4 = b·h³/3. This compact form is what appears in every mechanics textbook for a cantilever rectangle anchored at its base.
- What is the moment of inertia of a square?
- A square is a rectangle with b = h, so the formula simplifies to Ix = Iy = a⁴/12, where a is the side length. Polar moment J = Ix + Iy = a⁴/6. Section modulus S = I/c = a³/6. Plastic section modulus Z = a³/4. For a 100 mm square: Ix = 100⁴/12 = 8.33 × 10⁶ mm⁴.
- What is the moment of inertia of a hollow rectangle (box section)?
- For an outer rectangle b × h with a centred inner cavity bᵢ × hᵢ: Ix = (b·h³ − bᵢ·hᵢ³) / 12. The same logic applied to the perpendicular axis gives Iy = (h·b³ − hᵢ·bᵢ³)/12. The formula is just two rectangle formulas — outer minus inner — because moment of inertia is additive over disjoint areas. Used for HSS rectangular sections; the calculator above has a "hollow rectangle" mode that handles this directly.
- How does I scale with rectangle dimensions?
- Ix scales linearly with width b but with the cube of height h. Doubling height multiplies Ix by 8; doubling width only doubles it. This is why structural beams are rectangular tall (web vertical) rather than rectangular wide — for the same cross-sectional area, taller is much stiffer in bending. The same area arranged as a 100 × 200 vs 200 × 100 rectangle differs by a factor of 4 in Ix.
- How is the rectangle moment of inertia formula derived?
- Direct integration: Ix = ∫ y² dA, where y is the distance from the centroidal x-axis. For a rectangle centred on origin, y goes from −h/2 to +h/2 and dA = b·dy. So Ix = ∫−h/2+h/2 y² · b dy = b · [y³/3] from −h/2 to +h/2 = b · (2 · (h/2)³ / 3) = b·h³/12. Same approach with x integration gives Iy.
- What is the radius of gyration of a rectangle?
- rx = √(Ix/A) = √(b·h³/12 / (b·h)) = h/√12 ≈ 0.289·h. Similarly ry = b/√12. The radius of gyration is used in column slenderness ratio kL/r — taller (in the bending direction) means more buckling-resistant. For a 100 × 200 rectangle: rx = 200/√12 ≈ 57.7 mm, ry = 100/√12 ≈ 28.9 mm.
Sources and methodology
- Beer, F. P., Johnston, E. R., DeWolf, J. T., Mazurek, D. F. Mechanics of Materials, 7th Edition. McGraw-Hill, 2015. Chapter 9 — area moments of inertia of standard shapes.
- Young, W. C., Budynas, R. G. Roark's Formulas for Stress and Strain, 8th Edition. McGraw-Hill, 2011. Table 6.1, items 4 and 5 (rectangle, square).
- Crandall, S. H., Dahl, N. C., Lardner, T. J. An Introduction to the Mechanics of Solids, 2nd Edition. McGraw-Hill, 1978. Derivation of the parallel-axis theorem.
- American Institute of Steel Construction. Steel Construction Manual, 15th Edition. Table 1-11 — HSS rectangular section properties.