Pick a section shape and enter its dimensions to get area, centroid, and Ix / Iy about the centroidal axes.
About this tool
The Area Moment of Inertia Calculator is a free, in-browser tool that returns the section properties engineers need for bending and buckling: cross-sectional area A, centroid location, and the area moments of inertia Ix and Iy about the centroidal axes. Choose a rectangle, a solid circle, a hollow tube, or a doubly-symmetric I-beam and enter its dimensions in millimetres or inches. Everything is computed locally in your browser.
The standard formulas are used. A rectangle b×h has A = bh, Ix = bh³/12 and Iy = hb³/12. A solid circle of diameter d has A = πd²/4 and Ix = Iy = πd⁴/64. A hollow tube subtracts the bore: A = π(dₒ² − dᵢ²)/4 and I = π(dₒ⁴ − dᵢ⁴)/64. For the I-beam the section is treated as a full bounding rectangle minus the two web-side cut-outs: A = bf·H − (bf − tw)(H − 2tf), Ix = [bf·H³ − (bf − tw)(H − 2tf)³]/12, and Iy = [2·tf·bf³ + (H − 2tf)·tw³]/12, all about the centroid at mid-height. The tool also reports the section modulus Sx = Ix/(H/2).
As a worked example, a 50 mm × 100 mm rectangle (b = 50, h = 100) has A = 5,000 mm², a centroid 50 mm from the base, Ix = 50 × 100³/12 ≈ 4,166,667 mm⁴, Iy = 100 × 50³/12 ≈ 1,041,667 mm⁴, and Sx = Ix/50 ≈ 83,333 mm³. Ix is the strong-axis stiffness used in the deflection and stress formulas; Iy governs bending or buckling about the weak axis.
Frequently asked questions
What is the area moment of inertia?
The area (second) moment of inertia measures how a cross-section's area is distributed about a bending axis; larger values resist bending and buckling better. For a rectangle it is Ix = bh³/12, and for a solid circle it is I = πd⁴/64. Its units are length to the fourth power (mm⁴ or in⁴).
How is a hollow tube handled?
A hollow round tube subtracts the bore from the outer circle: area A = π(dₒ² − dᵢ²)/4 and I = π(dₒ⁴ − dᵢ⁴)/64, where dₒ is the outer and dᵢ the inner diameter. The inner diameter must be smaller than the outer, or the tool flags the input as invalid.
How is the I-beam computed?
It uses the doubly-symmetric method: a bounding rectangle bf×H minus the two web-side cut-outs. Ix = [bf·H³ − (bf − tw)(H − 2tf)³]/12 and Iy = [2·tf·bf³ + (H − 2tf)·tw³]/12, with H the overall height, bf the flange width, tf the flange thickness, and tw the web thickness.
Where is the centroid located?
For all four shapes here — rectangle, circle, tube, and doubly-symmetric I-beam — the section is symmetric, so the centroid sits at the geometric mid-height. The reported centroid distance is measured from the base to the neutral axis and equals half the overall height or diameter.