How to use this calculator

Pick a shape, enter its dimensions, and read the properties about the two centroidal axes. The x-axis is horizontal and the y-axis vertical, as drawn in the sketch. For an I-beam with a vertical web, x is the strong axis. The sketch redraws to scale as you type, which is a quick way to catch a mistyped dimension.

Once you have I_x, send it straight to the beam calculators with the link under the results.

Formulas

Shape	Area A	I_x	Plastic modulus Z_x
Rectangle b × h	b h	b h³ / 12	b h² / 4
Hollow rectangle B × H, wall t	B H − b_i h_i	(B H³ − b_i h_i³) / 12	(B H² − b_i h_i²) / 4
Solid circle, diameter d	π d² / 4	π d⁴ / 64	d³ / 6
Pipe, outside D, inside d_i	π (D² − d_i²) / 4	π (D⁴ − d_i⁴) / 64	(D³ − d_i³) / 6
I-beam d, b_f, t_f, t_w	2 b_f t_f + h_w t_w	[b_f d³ − (b_f − t_w) h_w³] / 12	b_f t_f (d − t_f) + t_w h_w² / 4

Here b_i = B − 2t and h_i = H − 2t are the inside dimensions of the hollow rectangle, d_i = D − 2t for the pipe, and h_w = d − 2t_f is the clear web height of the I-beam. The y-axis formulas swap the roles of width and height.

Derived propertiesS = Ic r = √I / A J_circle = I_x + I_y = π d⁴32

c is the distance from the centroid to the extreme fiber (half the depth for these symmetric shapes). The polar moment J equals the torsion constant only for circular sections, so the calculator reports J for circles and pipes only.

Worked example

Rectangle 100 mm wide × 200 mm deep (the default)

A = 100 × 200 = 20,000 mm².
I_x = 100 × 200³ / 12 = 6.667 × 10⁷ mm⁴. Turning the same board flat gives I_y = 200 × 100³ / 12 = 1.667 × 10⁷ mm⁴, one quarter as much.
S_x = I_x / (h/2) = 6.667 × 10⁷ / 100 = 666,700 mm³; Z_x = 100 × 200² / 4 = 1,000,000 mm³, so the shape factor Z/S is 1.5, as it always is for a rectangle.
r_x = √(I_x/A) = 57.7 mm.

Second moment of area, polar moment, or mass moment of inertia?

Three different quantities share the name “moment of inertia”:

Second moment of area I (m⁴, in⁴) measures how a cross-section resists bending. This is what beam deflection and stress formulas need, and what this calculator returns.
Polar moment of area J (m⁴, in⁴) measures resistance to twisting of circular shafts.
Mass moment of inertia (kg·m², lb·ft²) measures resistance to angular acceleration in dynamics. It is a different quantity with different units.

Elastic versus plastic section modulus

The elastic modulus S gives the moment at first yield, M_y = σ_y S. The plastic modulus Z gives the moment when the whole section has yielded, M_p = σ_y Z. Steel design codes such as AISC 360 base the bending capacity of compact sections on Z.

Assumptions and limits

All shapes are doubly symmetric, so the centroid and the plastic neutral axis are at the geometric center.
Corners are sharp. Rolled I-beams have fillets between web and flange and cold-formed tubes have rounded corners, so published tables differ by a few percent. With the default US I-beam (close to a W8×31), this calculator gives I_x ≈ 108 in⁴ against 110 in⁴ in the AISC Manual.
Use manufacturer or code tables for final design of standard sections.

Frequently asked questions

Which axis should I use for a beam?

The axis the beam bends about. A joist loaded from above with its tall side vertical bends about the horizontal x-axis, so use I_x. For column buckling, use the smaller of I_x and I_y unless the column is braced in the weak direction.

How do I handle a shape that is not listed, like a T or channel?

Split it into rectangles, find the overall centroid, and add each rectangle’s own I plus its area times the squared distance to the centroid (the parallel axis theorem): I = Σ (I_i + A_i d_i²).

Why do my units look huge in mm⁴?

Fourth powers grow quickly. 1 cm⁴ = 10,000 mm⁴ and 1 m⁴ = 10¹² mm⁴. Use the unit menu next to each result to switch to cm⁴, m⁴ or in⁴.

For education and preliminary checks only. Have a qualified engineer verify any design before it is built.