Area Moment of Inertia Calculator

Every beam, column, and shaft resists bending and twisting according to how its cross-section is shaped — not just how much material it contains. Two beams with the identical cross-sectional area can have very different stiffness depending on whether that material sits close to the bending axis or far from it. The area moment of inertia (also called the second moment of area) is the number engineers use to capture that shape effect.

This calculator finds the area moment of inertia, section modulus, and radius of gyration for six common cross-sections: solid rectangle, hollow rectangle (tube), circle, round tube (pipe), isosceles triangle, and symmetric I-beam. It covers area moment of inertia calculation, second moment of area, section modulus, and radius of gyration in one tool, for the shapes most often needed in beam, column, and shaft design.

What This Area Moment of Inertia Calculator Does

Pick a cross-section shape, enter its dimensions in any consistent length unit, and the calculator returns the section properties that describe how that shape resists bending and twisting.

Six Common Cross-Sections

Solid rectangle and hollow rectangle (tube) cover typical timber, plate, and rectangular structural tubing sections. Circle and round tube cover solid rods, shafts, and pipe. Isosceles triangle covers triangular bracing and gusset shapes. Symmetric I-beam covers a doubly-symmetric wide-flange shape idealized as three rectangles — two equal flanges and a web.

What You Get Back: Ix, Iy, Section Modulus, and Radius of Gyration

For each shape the calculator reports the cross-sectional area, the moment of inertia about the horizontal centroidal axis (Ix) and vertical centroidal axis (Iy), the section modulus (Sx and, where relevant, Sy), and the radius of gyration (rx and, where relevant, ry). For circular shapes it also reports the polar moment of inertia, J.

What Is the Area Moment of Inertia (Second Moment of Area)

The area moment of inertia measures how a cross-section’s area is distributed relative to a given axis. Material located far from the axis contributes much more to the moment of inertia than the same amount of material located close to the axis, because the contribution scales with the square of the distance.

Why It Matters for Bending and Stiffness

A beam’s resistance to bending deflection is proportional to its moment of inertia about the axis it is bending around — this is the “I” in the beam deflection and bending stress equations used throughout structural and mechanical engineering. A larger Ix means a stiffer beam for bending about the x-axis, which is why structural shapes like I-beams put most of their material in flanges far from the centroidal axis rather than spread out near it.

Moment of Inertia vs Mass Moment of Inertia

The area moment of inertia (this calculator) is a purely geometric property of a cross-section’s shape, used for bending and torsion of beams and shafts. It is a different quantity from the mass moment of inertia, which measures a body’s resistance to rotational acceleration and depends on mass distribution rather than cross-sectional area. The two share a name and similar-looking formulas for some shapes, which is a common source of confusion, but they answer different engineering questions and are not interchangeable.

Moment of Inertia for a Solid Rectangle

Formula:

Ix = (b × h³) ÷ 12
Iy = (h × b³) ÷ 12

Where:

b = width
h = height (the dimension in the direction of bending for Ix)

Worked Example

For a rectangle 4 in wide and 6 in tall:

Ix = (4 × 6³) ÷ 12 = (4 × 216) ÷ 12 = 72 in&sup4;
Iy = (6 × 4³) ÷ 12 = (6 × 64) ÷ 12 = 32 in&sup4;

Why Height Matters More Than Width

Because h is cubed in the Ix formula, doubling a rectangle’s height increases Ix eightfold, while doubling its width only doubles Ix. This is why joists and beams are oriented with their long dimension vertical — standing a board on edge makes it dramatically stiffer against bending than laying it flat, even though the cross-sectional area is identical either way.

Moment of Inertia for a Hollow Rectangle (Tube)

This calculator asks for the outer width and height plus a uniform wall thickness, matching how rectangular structural tubing is typically specified, then computes the inner dimensions and subtracts the inner rectangle’s moment of inertia from the outer one.

Formula:

Ix = (B × H³ − b × h³) ÷ 12
Iy = (H × B³ − h × b³) ÷ 12

Where:

B, H = outer width and height
b, h = inner width and height (outer dimensions minus twice the wall thickness)

Worked Example

For a tube with outer width 10 in, outer height 8 in, and a 2 in wall thickness (inner width 6 in, inner height 4 in):

Ix = (10 × 8³ − 6 × 4³) ÷ 12 = (5,120 − 384) ÷ 12 = 394.7 in&sup4;

Moment of Inertia for a Circle and Round Tube

Circular sections are the one case where Ix and Iy are always equal, since the shape is symmetric about every axis through its center.

Formula (solid circle):

Ix = Iy = (π × D⁴) ÷ 64

Formula (round tube):

Ix = Iy = π × (D⁴ − d⁴) ÷ 64

Where D is the outer diameter and d is the inner diameter (this calculator asks for outer diameter and wall thickness, and computes the inner diameter from those).

Worked Example

For a solid circle 10 in in diameter:

Ix = (π × 10⁴) ÷ 64 = (π × 10,000) ÷ 64 ≈ 490.9 in&sup4;

Polar Moment of Inertia and Torsion

For a circular or round-tube cross-section, the polar moment of inertia J = Ix + Iy describes resistance to twisting (torsion) about the shaft’s own axis, and this calculator reports it for those two shapes. For non-circular shapes — rectangles, triangles, and I-beams — the true torsional resistance depends on more than just Ix + Iy and requires a separate, shape-specific torsion analysis, so this calculator does not report a torsion constant for those shapes.

Moment of Inertia for an Isosceles Triangle

Formula (about the horizontal centroidal axis, located one-third of the height up from the base):

Ix = (b × h³) ÷ 36

Formula (about the vertical centroidal axis, assuming the apex sits directly above the midpoint of the base):

Iy = (b³ × h) ÷ 48

Worked Example

For a triangle with a 6 in base and 9 in height:

Ix = (6 × 9³) ÷ 36 = (6 × 729) ÷ 36 = 121.5 in&sup4;
Iy = (6³ × 9) ÷ 48 = (216 × 9) ÷ 48 = 40.5 in&sup4;

Why This Calculator Assumes an Isosceles Shape

The Iy formula above only holds when the triangle is symmetric about a vertical axis through the midpoint of its base — an isosceles triangle. A scalene triangle’s moment of inertia depends on the exact position of all three vertices, not just a base and height, so this calculator does not attempt to compute Iy for a general triangle. Ix about the horizontal centroidal axis, by contrast, only depends on base and height regardless of the triangle’s exact shape, which is why it is reported for any triangle sharing that base and height.

Moment of Inertia for a Symmetric I-Beam

This calculator idealizes an I-beam as three rectangles — two equal flanges and a web — with no fillets or rounded corners, which is standard for a quick estimate but will differ slightly from a real rolled shape’s published catalog values.

Formula:

Ix = [B × H³ − (B − tw) × (H − 2 × tf)³] ÷ 12
Iy = [2 × tf × B³ + (H − 2 × tf) × tw³] ÷ 12

Where:

H = total height
B = flange width
tf = flange thickness
tw = web thickness

Worked Example

For a beam 12 in tall, with 6 in wide flanges, 1 in flange thickness, and a 0.5 in web:

Ix = [6 × 12³ − 5.5 × 10³] ÷ 12 = [10,368 − 5,500] ÷ 12 ≈ 405.7 in&sup4;
Iy = [2 × 1 × 6³ + 10 × 0.5³] ÷ 12 = [432 + 1.25] ÷ 12 ≈ 36.1 in&sup4;

Why Real Rolled Shapes Differ Slightly

A manufactured steel W-shape or I-beam has fillets where the web meets the flanges and slightly tapered flange faces, both of which add a small amount of extra material and shift the true properties away from the idealized three-rectangle formula above. For a specific catalog shape, use the manufacturer’s or standard’s published section properties rather than this calculator’s idealized estimate.

Section Modulus and Radius of Gyration Explained

Section Modulus and Bending Stress

Section modulus (S = I ÷ c, where c is the distance from the centroidal axis to the extreme fiber) converts the moment of inertia into a value used directly in bending stress calculations: a larger section modulus means lower bending stress for the same applied bending moment.

Radius of Gyration and Column Buckling

Radius of gyration (r = the square root of I divided by area) is a measure of how efficiently a cross-section’s area is distributed to resist buckling, and it is a key input to column buckling calculations, where a smaller radius of gyration about a given axis means the column is more prone to buckling about that axis.

Area Moment of Inertia Calculator vs Other Tools on This Site

This site’s Cross-Sectional Area Calculator reports area and perimeter for duct, pipe, and tube cross-sections — useful for sizing, flow, and material takeoffs. This calculator instead reports the section properties — moment of inertia, section modulus, and radius of gyration — used for structural and mechanical stiffness, bending stress, and buckling calculations. Use the Cross-Sectional Area Calculator when you need area or perimeter, and this calculator when you need stiffness-related section properties for the same general shapes.

Frequently Asked Questions

What is the difference between area moment of inertia and mass moment of inertia?

Area moment of inertia is a geometric property of a cross-section’s shape, used for bending and torsion of beams and shafts. Mass moment of inertia measures a body’s resistance to rotational acceleration and depends on how its mass is distributed, not its cross-sectional shape. They serve different engineering calculations despite the similar name.

Why is Ix different from Iy for the same shape?

Ix and Iy measure resistance to bending about two different axes, and unless a shape is symmetric in both directions (like a circle or a square), material is distributed differently relative to each axis, giving different values. A rectangle standing on its narrow edge has a much larger Ix than Iy, for example.

Does this calculator account for fillets on rolled steel I-beams?

No. It idealizes the I-beam as three flat rectangles with sharp corners. Real rolled shapes have fillets where the web meets the flanges, which add a small amount of material and shift the true properties slightly. For a specific catalog shape, use its published section properties instead.

Why doesn’t this calculator report a torsion constant for rectangles or I-beams?

The polar moment of inertia (Ix + Iy) only equals the actual torsion constant for circular and round-tube cross-sections. For rectangular, triangular, and I-beam shapes, torsional resistance depends on the shape in a more complex way that requires a separate, shape-specific torsion analysis, so this calculator reports Ix and Iy for those shapes but does not label a torsion constant for them.

What units does this calculator use?

Whatever consistent length unit you enter your dimensions in. If you enter inches, area comes out in in², moment of inertia in in&sup4;, and section modulus in in³. Switching the unit label does not convert your numbers — it only relabels the results to match whatever unit you are actually entering.

Can I use this for a scalene (non-isosceles) triangle?

Ix about the horizontal centroidal axis works for any triangle sharing the same base and height. Iy assumes an isosceles triangle — apex centered above the midpoint of the base — because a scalene triangle’s Iy depends on the exact position of all three vertices, not just base and height.

How is radius of gyration used in practice?

It is a required input to column buckling formulas (such as the slenderness ratio, length divided by radius of gyration), where a column is checked for buckling about the axis with the smaller radius of gyration, since that is typically the weaker direction.