Moment of Inertia Calculator

Area moment of inertia (Iₓ, I_y), centroid, section modulus and radius of gyration for 9 cross-sections — rectangles, circles, I-beams, T/L sections and standard AISC W-shapes.

Cross-section

Inputs change to match the selected shape. All math referenced to centroidal axes.
in
in

Section properties

Iₓ — strong axis

— in⁴

I_y — weak axis

— in⁴

Area (A)

— in²

Polar moment (J = Iₓ + I_y)

— in⁴

Section modulus Sₓ

— in³

Section modulus S_y

— in³

Radius of gyration rₓ

— in

Radius of gyration r_y

— in

Strong vs weak axis: Iₓ is bending about the horizontal axis (vertical beam orientation); I_y is bending about the vertical axis. For asymmetric T and L sections the centroid is offset from the geometric centre — use the parallel-axis theorem when transforming to a different axis.

Area moment of inertia (units in⁴ or mm⁴), not mass moment of inertia (units kg·m²). These are different physical quantities — area MoI is used for bending and deflection of beams; mass MoI is used for rotational dynamics.

Why moment of inertia matters in structural design

The area moment of inertia (I) is the geometric property that quantifies how the cross-section resists bending. A larger I means less deflection under load, less bending stress at the extreme fiber, and a higher critical buckling load.

deflection: δ ∝ 1 / I  ·  bending stress: σ = M·y / I  ·  buckling: P_cr = π² E I / L²

A 2×8 floor joist has Iₓ ≈ 47.6 in⁴ (strong axis, on edge) vs 5.4 in⁴ on the flat — 9× stiffer simply by orienting the cross-section to put material away from the neutral axis. This is why beams are tall and narrow, and why I-beams are dramatically more efficient than rectangular bars of the same area.

Section modulus and radius of gyration

Section modulus S = I / c where c is the distance from the centroid to the extreme fiber. Allowable bending moment is M = S × Fb, so S is what you compare against the design moment in a flexural check.

Radius of gyration r = √(I / A) is the effective radius at which the area could be concentrated to give the same moment of inertia. Slenderness ratio (KL / r) determines buckling capacity of compression members.

Cross-section formulas

ShapeIₓA
Solid Rectangle (b × h)b·h³ / 12b·h
Hollow Rectangle (B,H outer; b,h inner)(B·H³ − b·h³) / 12B·H − b·h
Solid Circle (Ø D)π·D⁴ / 64π·D² / 4
Hollow Circle (D outer, d inner)π·(D⁴ − d⁴) / 64π·(D² − d²) / 4
I-Beam(bf·H³ − (bf−tw)·hw³) / 122·bf·tf + hw·tw
Triangle (b base, h height)b·h³ / 36 (centroid)b·h / 2

T and L sections require first computing the centroid, then applying the parallel-axis theorem to each rectangular sub-part — the calculator does this automatically.

Worked examples

Solid Rectangle 4 × 8 in (strong axis vertical)

  • Iₓ = (4 × 8³) / 12 = 170.67 in⁴
  • I_y = (8 × 4³) / 12 = 42.67 in⁴
  • A = 4 × 8 = 32 in²
  • Iₓ / I_y = 4× — meaningful but not as dramatic as an I-beam

Solid Circle Ø 4 in

  • I = π × 4⁴ / 64 = 12.566 in⁴
  • A = π × 4² / 4 = 12.566 in²
  • r = √(12.566 / 12.566) = 1.0 in
  • Iₓ = I_y by symmetry; J = 2·I = 25.13 in⁴

AISC W8×10 (lookup)

  • A = 2.96 in²
  • Iₓ = 30.8 in⁴ (strong axis)
  • I_y = 2.09 in⁴ (weak axis)
  • Sₓ = 7.81 in³
  • Iₓ / I_y ≈ 15× — the value of the I-beam shape

Frequently asked questions

What is the moment of inertia?
In structural design it refers to the area moment of inertia (second moment of area) — the geometric property that quantifies a cross-section's resistance to bending about a given axis. Units are length⁴ (in⁴ or mm⁴). It is NOT mass moment of inertia (kg·m²), which applies to rotating bodies.
What is the difference between Iₓ and I_y?
Iₓ is the moment of inertia about the horizontal (x) centroidal axis — used for bending in the vertical plane. I_y is about the vertical (y) axis. For typical beam orientations, Iₓ is the "strong axis" and is larger. I-beams have Iₓ that is 5-20× larger than I_y, which is why they are installed with the web vertical.
What is the moment of inertia of a 2×8?
A nominal 2×8 measures 1.5″ × 7.25″ actual. Iₓ (about strong axis, on edge) = (1.5 × 7.25³) / 12 = 47.63 in⁴. I_y (on the flat) = (7.25 × 1.5³) / 12 = 2.04 in⁴. The on-edge orientation is 23× stiffer.
How does moment of inertia affect deflection?
Beam deflection is inversely proportional to I: δ = (5·w·L⁴) / (384·E·I) for a uniformly loaded simply-supported beam. Doubling I halves the deflection. This is why we use deep, thin sections for floor joists — they put material far from the neutral axis to maximise I.
What is the section modulus?
Section modulus S = I / c where c is the distance from the centroid to the extreme fiber. Maximum bending stress is σ = M / S, so S is the property you compare directly against the allowable bending stress F_b in a code check. Units are length³ (in³ or mm³).
What is the parallel axis theorem?
For an axis parallel to the centroidal axis at distance d, the moment of inertia is I_axis = I_centroidal + A·d². This is essential for composite shapes — split into simple sub-shapes, compute I about each sub-shape's centroid, then translate to the composite centroid using the theorem.
How do I find the moment of inertia of a W-beam?
Look it up in the AISC Steel Construction Manual — Iₓ, I_y, Sₓ, S_y, rₓ and r_y are tabulated for every standard W-shape. Our calculator includes 15 common W4 through W24 sections from the 15th edition.