Free Moment of Inertia Calculator

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The (area) moment of inertia — sometimes called the second moment of area — measures how a cross-section's area is distributed relative to a reference axis, specifically the centroidal axis here. It's one of the most important quantities in structural and mechanical engineering: a beam's resistance to bending stress and its deflection under load both depend directly on this value, which is exactly why steel I-beams are shaped the way they are (pushing material away from the centroidal axis dramatically increases I without adding much weight).

How it works

Choose a shape and enter its dimensions. For a rectangle with base b and height h, I=bh³/12 about the axis through its centroid parallel to the base. For a circle of radius r, I=πr⁴/4. For a triangle with base b and height h, I=bh³/36 about the horizontal axis through its centroid. Each formula comes from integrating y² over the shape's area, where y is distance from the centroidal axis — shapes with more area concentrated far from that axis end up with a larger I.

Enter shape.
Enter base (rectangle / triangle).
Enter height (rectangle / triangle).
Enter radius (circle).
Click Calculate to see your results.

Examples

Rectangle, b=4, h=6

I = 4(6³)/12 = 4(216)/12 = 72.

Circle, r=3

I = π(3⁴)/4 = π(81)/4 ≈ 63.62.

Triangle, b=6, h=9

I = 6(9³)/36 = 6(729)/36 = 121.5.

Common mistakes to avoid

Using the wrong axis reference — these formulas are specifically for the centroidal axis, not the base or any other edge (the parallel axis theorem is needed to shift to a different axis).
Mixing up the area moment of inertia (used for bending stress) with the mass moment of inertia (used for rotational dynamics) — they measure different physical quantities.
Applying the rectangle formula's "12" or the triangle formula's "36" to the wrong shape.

Frequently asked questions

The triangle's centroid sits at 1/3 of its height from the base rather than at mid-height (like a rectangle's centroid), which changes the constant that falls out of the integration — 36 instead of 12.

A larger I means the shape resists bending more strongly about that axis — for the same material and span, a beam with a higher I will deflect less under the same load.

No — that's a related but distinct concept measuring how mass (not area) is distributed for rotational motion (units of mass×length²). This calculator computes the area moment of inertia (units of length⁴), used in beam bending, not rotational dynamics.

By symmetry, a circle's moment of inertia is the same about any axis through its center, so there's no separate "base" dimension to account for — it depends only on the radius.

Because it weights each small piece of area by the square of its distance from the axis, and integrating that over a shape with a linear dimension like height or radius naturally produces a higher power (cubed for rectangles/triangles, to the fourth for a circle) than plain area alone.