Moment of Inertia Calculator

What it calculates

This free online moment of inertia calculator returns the full set of section properties for a cross-section: area, centroid, area moment of inertia (Ix and Iy, also written Ixx and Iyy), elastic section modulus, plastic section modulus, radius of gyration, polar moment of inertia and the principal axes. No login and no download - the whole calculation runs in your browser.

Nine shapes are built in: solid rectangle, hollow rectangle, solid circle, hollow circle (tube), I-section, channel, tee, triangle, and an arbitrary polygon you can draw point by point. Every result is reported about both principal axes, in mm, cm or inches.

What you enter

Shape

One of nine: rectangle, hollow rectangle, circle, tube, I-section, channel, tee, triangle, custom polygon

Dimensions

The shape's own dimensions - width, height, flange and web thickness, root radius, diameter

Unit

mm, cm or inch; results convert with the input

For a custom polygon you enter the vertex coordinates directly, or drag points on the canvas. No material properties are needed - every quantity here is purely geometric.

What you get back

Area and centroid position, Ix and Iy about the centroidal axes, the product of inertia Ixy, elastic section modulus to each extreme fibre (top, bottom, left, right - they differ for an unsymmetric section), plastic section modulus, radius of gyration about both axes, polar moment of inertia, and the principal moments with the principal-axis angle.

Each result comes with a step-by-step derivation showing the substitution, not just the answer, and exports to Excel (.xlsx) and PDF.

Section modulus - elastic

The elastic section modulus is S = I / c, where c is the distance from the centroid to the extreme fibre. It converts a bending moment into a peak stress: sigma = M / S. Because c differs top from bottom on an unsymmetric section, this calculator reports the elastic section modulus separately for the top, bottom, left and right fibres rather than a single value - the smaller one governs, since that fibre reaches yield first.

For the 100 x 200 mm rectangle the tool opens with, S = 666 667 mm3 (666.7 cm3) about the strong axis. A 50 kNm moment on that section gives a peak bending stress of 50e6 / 666667 = 75.0 N/mm2.

Section modulus - plastic

The plastic section modulus Z is the first moment of area of the two halves about the plastic neutral axis, and it governs the fully-plastic moment Mpl = Z x fy used in plastic design and in EN 1993 cross-section classes 1 and 2. The ratio Z / S is the shape factor - how much moment a section carries beyond first yield.

The shape factor is a useful sanity check on any section-property tool: for a solid rectangle it is exactly 1.5, and this calculator returns Z = 1 000 000 mm3 against S = 666 667 mm3 for the default rectangle, giving 1.500. A typical I-section is far lower - the worked example below comes out at 1.134 - because an I-section already concentrates its area at the extreme fibres, so there is less reserve to recover.

Radius of gyration

The radius of gyration i = sqrt(I / A) is the distance from the axis at which the entire area could be concentrated without changing the moment of inertia. It is the property that drives column buckling: the slenderness ratio is the effective length divided by the radius of gyration, and a column always buckles about the axis with the smaller value.

For the default 100 x 200 mm rectangle the calculator returns 57.74 mm about the strong axis and 28.87 mm about the weak axis - a factor of two, which is why a rectangular column of this proportion buckles about the weak axis at a quarter of the load.

Polar moment of inertia

The polar second moment of area is the sum of the two in-plane moments of inertia, Ip = Ix + Iy, by the perpendicular-axis theorem. It measures resistance to twisting about the longitudinal axis, and for a solid circle it reduces to pi x d^4 / 32.

One caution that matters in practice: for a non-circular section the polar second moment of area is NOT the torsion constant. The torsional stiffness GJ uses J, which for an open section such as an I or a channel is very much smaller than Ix + Iy. This calculator reports the polar second moment of area Ip = Ix + Iy, which is exact for circular sections and should not be substituted for J on an open section.

Custom shapes and polygons

For a cross-section that is not one of the eight standard shapes, the polygon mode takes an arbitrary closed outline - any number of vertices, convex or concave - and integrates the section properties over it using the shoelace formulation. This covers built-up plate girders, notched sections, cranked plates and any hand-drawn profile.

A polygon section is generally unsymmetric, so the product of inertia Ixy is non-zero and the principal axes are rotated away from the input axes. The tool reports Ixy, the two principal moments and the principal-axis angle, so you can see how far the strong axis is rotated from horizontal - a detail that is easy to miss and changes the bending stress substantially.

Worked example - I-section 150 x 300, tf 12, tw 8

Take a welded I-section: 150 mm wide flanges 12 mm thick, 300 mm overall depth, 8 mm web, no root fillet. These are the I-section defaults in the tool, so every figure below is reproducible by selecting I-section and pressing nothing else.

Web clear height, h - 2 tf

300 - 24 = 276 mm

Area A

2 x (150 x 12) + 276 x 8 = 5808 mm2

Ix (strong axis)

88 709 184 mm4 = 8871 cm4

Iy (weak axis)

6 761 776 mm4 = 676 cm4

Elastic modulus Sx = Ix / (h/2)

591 395 mm3 = 591.4 cm3

Plastic modulus Zx

670 752 mm3 = 670.8 cm3

Shape factor Zx / Sx

1.134

Radius of gyration, strong axis

123.59 mm

Radius of gyration, weak axis

34.12 mm

Polar moment Ip = Ix + Iy

95 470 960 mm4 = 9547 cm4

The two flanges alone contribute 74 649 600 mm4 to Ix - that is 84.2 per cent of the total, from 62 per cent of the area. This is the whole reason I-sections exist: material at the extreme fibre earns its inertia by the square of its distance from the centroid, so moving area outward is far more effective than adding it near the neutral axis.

Note also Ix / Iy = 13.1. The section is thirteen times stiffer one way than the other, which is why an unrestrained I-beam needs a lateral-torsional buckling check rather than a simple bending check.

Formula reference

Rectangle, strong axis

$I = \dfrac{b\,h^{3}}{12}$

Solid circle

$I = \dfrac{\pi\,d^{4}}{64}$

Hollow circle

$I = \dfrac{\pi\,(D^{4} - d^{4})}{64}$

Parallel-axis shift

$I = I_c + A\,d^{2}$

Elastic section modulus

$S = \dfrac{I}{c}$

Radius of gyration

$i = \sqrt{\dfrac{I}{A}}$

Polar second moment

$I_p = I_x + I_y$

Bending stress

$\sigma = \dfrac{M}{S}$

Built-up sections (I, channel, tee) are split into rectangles, each computed about its own centroid and then shifted to the overall centroid by the parallel-axis theorem before summing. The theory page derives all of this and covers where each formula comes from.

Assumptions and limits

Every quantity here is purely geometric. No material property, no load and no code check is involved, so the results are the same whatever steel, concrete or timber the section is made from - which also means nothing here tells you whether a section is adequate.

Sections are treated as thin-walled where the formula does (the derivation is exact for the rectangles a built-up section is decomposed into) and as fully effective throughout - there is no allowance for local buckling or for the effective-width reduction that a slender class-4 cross-section needs under EN 1993-1-5. Holes, notches and bolt deductions are not included unless you model them in the polygon outline.

The plastic modulus assumes a fully plastic stress block and is only meaningful for a cross-section that can reach it: class 1 and 2 under EN 1993. Torsion is limited to the polar second moment of area as noted above, and shear-area properties and warping constants are outside the scope of this tool.

Frequently asked questions (FAQ)