travis·a·walters

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Area Moments of Inertia

Live calculators for common cross sections, with the formulas behind them. Every property is taken about axes through the centroid; enter dimensions in any consistent unit.

A
cross-sectional area · unit2
Ix, Iy
second moment of area about the centroidal x / y axis; resists bending · unit4
Sx, Sy
section modulus I/c, where c is the distance to the extreme fiber; bending stress σ = M/S · unit3
J
polar moment of area; resists torsion · unit4
click any result to copy

Rectangle

x y b h
A =bh
Ix = bh312
Iy = hb312
J = bh(b2 + h2)12

Hollow Rectangle

x y B H b × h
A =BH − bh
Ix = BH3 − bh312
Iy = HB3 − hb312

Outer dimensions B × H, inner hole b × h, both centered on the centroid.

Circle

x y r
A =πr2
Ix = Iy = πr44
J = πr42

Hollow Circle

x y R r
A =π(R2 − r2)
Ix = Iy = π(R4 − r4)4
J = π(R4 − r4)2

Triangle

x y b h
A = bh2
Ix = bh336

Centroid sits at h/3 above the base. Ix is taken about the centroidal axis parallel to the base. Sx uses the apex — the farther fiber, 2h/3 from the centroid, which governs the bending stress.

Semicircle

x y r
A = πr22
Ix = (9π2 − 64)r472π ≈ 0.1098 r4
Iy = πr48

Centroid sits 4r/(3π) ≈ 0.424r above the flat edge. Ix is about the centroidal axis parallel to the flat edge; Iy is about the axis of symmetry. Sx uses the crown of the arc — the farther fiber from the centroid.

I-Beam (Symmetric)

x y bf d tf tw
A =2bftf + tw(d − 2tf)
Ix = bfd3 − (bf − tw)(d − 2tf)312
Iy = 2tfbf3 + (d − 2tf)tw312

Flange width bf, flange thickness tf, web thickness tw, overall depth d. Assumes equal top and bottom flanges. Real rolled sections have fillet radii at flange-web transitions; use published section property tables for code work.