Moment of Inertia Calculator

The phrase moment of inertia covers two different quantities, and most confusion comes from mixing them. The second moment of area, also called the area moment of inertia or the 2nd moment of inertia, describes a cross-section. It tells you how well a beam's cross-section resists bending. Its units are length to the fourth power, such as mm4 or in4, and it depends only on the shape of the cross-section. The mass moment of inertia, also called rotational inertia, describes how hard it is to change the spin of a solid object. Its units are mass times length squared, such as kg m2 or lb ft2, and it depends on the mass and how that mass is spread around the axis.

This calculator keeps the two apart. The Area modes at the top of the shape list return second moment of area, section modulus and related section properties. The Mass modes return rotational inertia in kg m2, lb ft2, g cm2 and lb in2. The comparison table on this page lists what each quantity is used for. If you are checking a beam or a bracket cross-section, use the area modes. If you are working on a flywheel, wheel, rod or any spinning part, use the mass modes.

Both share one idea: material far from the axis counts more than material near it, because the distance is squared. That one fact explains why deep beams are stiff, why I-beams are shaped the way they are, and why a ring is harder to spin than a solid disk of the same mass.

For a rectangle of width b and height h, the second moment of area about the horizontal axis through the centroid is Ix = b h³ / 12. The height is cubed, so depth is far more valuable than width. Doubling the depth gives eight times the Ix, while doubling the width only doubles it. The 100 × 200 mm example gives Ix = 6,666.7 cm4, but turned on its side with 200 mm as the width, it gives only 1,666.7 cm4.

For a solid circle of diameter d, I = pi d⁴ / 64, the same about any axis through the centre. The fourth power makes small increases in diameter matter: a 100 mm bar has sixteen times the I of a 50 mm bar. A hollow circle, such as a pipe or tube, is the solid outer circle minus the empty inner circle: I = pi (D⁴ − d⁴) / 64. The tube in the examples keeps about 52 percent of a solid bar's I with a much smaller share of the material.

For a triangle with base b and height h, the centroid sits one third of the height above the base, and Ix = b h³ / 36 about the centroidal axis. About the base it is b h³ / 12. The calculator reports Iy = h b³ / 48, which applies to an isosceles triangle with the apex above the middle of the base. These formulas are standard results, and the Wikipedia articles on the second moment of area and on the list of moments of inertia give the derivations and more shapes.

Real sections such as I-beams, channels and tees are built from rectangles. Two methods give the same answer. The subtraction method treats an I-beam as the full outer rectangle minus the two side pockets: Ix = [bf h³ − (bf - tw)(h - 2 tf)³] / 12. The composite method adds each part's own I plus its area times the squared distance to the common axis. For the I-beam example both give 1,735.0 cm4.

That second method is the parallel axis theorem: I = Ic + A d^2. It is the tool for any T-beam, L-section, box section or irregular shape you can divide into simple parts. First find the centroid of the whole shape by area-weighting the parts. Then, for each part, add its own centroidal I and its area times the squared distance from its centroid to the whole shape's centroid. Holes are subtracted. The T-section mode does this for a tee and reports the centroid height; the shift mode lets you apply the theorem to any part you have worked out yourself.

If you work in a spreadsheet, set up one row per part with columns for width, height, area, distance and I, then sum. The sheet will match this page. For an outline that cannot be divided into simple parts, use CAD software or the polygon method described in the Wikipedia article on the second moment of area. The table of I-beam against solid bar shows the payoff of the shape: the same cross-section area as a solid bar, with about 33 times the Ix.

The calculator reports a few related section properties alongside Ix and Iy. The section modulus Zx = Ix / c divides Ix by c, the distance from the axis to the farthest edge, and links a bending moment to the peak bending stress. For the 100 × 200 mm rectangle c is 100 mm and Zx = 666.7 cm3. The radius of gyration rx = √(Ix / A) is a single distance that describes how spread out the area is; for that rectangle it is 57.7 mm. The polar moment J = Ix + Iy describes a section's geometric resistance to twisting about the centroid.

These numbers are section properties. They are inputs to a structural calculation, not the calculation itself. In structural engineering the second moment of area is an input to the bending stress and deflection of a beam, and a larger value means a stiffer section for the same material, span and load. A real check also needs the load, span, support conditions, material properties, safety factors and the rules in the applicable code.

This page does not do that check, and none of its numbers are a structural design. Treat them as estimates, use published section tables for real products, follow local building codes and the manufacturer's data, and have a qualified structural engineer check any work where failure could hurt someone or cost real money.

A spinning object resists a change of rotation in proportion to its mass moment of inertia, so the same push produces less angular speed-up for a larger I. For a point mass, I = m r^2. For an extended object, add up m r² over all its parts, which gives the standard results listed in the table of mass formulas: a solid cylinder or disk about its axis is m r² / 2, a solid sphere is 2 m r² / 5, a thin spherical shell is 2 m r² / 3, a thin ring is m r², and a slender rod is m L² / 12 about its centre or m L² / 3 about one end. These match the OpenStax University Physics text and the Wikipedia list of moments of inertia.

The comparison table shows how shape matters at equal mass and radius. A 10 kg object of radius 0.1 m has I = 0.040 kg m2 as a solid sphere, 0.050 as a solid cylinder, 0.067 as a thin shell and 0.100 as a thin ring. The ring is 2.5 times the sphere because all its mass sits at the full radius.

Axis choice matters too. A cylinder has one value about its own axis and another about a transverse axis through the centre, and the calculator returns both. A rod has one value at the centre and four times as much at the end. For three-dimensional bodies not on the list, split them into simple solids and combine them with the parallel axis theorem.

The parallel axis theorem lets you move from the axis through the centroid or centre of mass to any parallel axis. For area, I = Ic + A d^2. For mass, I = Icm + m d^2. The shift is always an increase, and it grows with the square of the distance. For the 100 × 200 mm rectangle (Ic = 6,666.7 cm4, A = 200 cm2), an axis at the base, 100 mm away, gives 26,666.7 cm4, which is b h³ / 3, four times the centroidal value. Use the shift modes when you have worked out Ic and A yourself, for example for a part of a composite section.

Units need care. Second moment of area scales with the fourth power of the length unit, so 1 in4 is 41.623 cm4 and 1 cm4 is 10,000 mm4. Mass moment of inertia scales with mass times the square of the length, so 1 kg m2 is 23.730 lb ft2. Pick a length unit for each shape once, enter every dimension in it, and read the result in whichever unit you need. The unit table on this page lists the common factors.

For the shift modes, the entered I, area and distance must use the same length unit (and mass unit for the mass version). The most common error is entering a cm4 value while the length unit is set to mm.

Choose a shape from the list. The Area modes ask for the dimensions of the cross-section: width and height for a rectangle, diameters for circles, flange, web and depth for an I-beam or tee. The Mass modes ask for the mass and size of the object. The default values give the 100 × 200 mm rectangle, so you see 6,666.7 cm4 straight away. Results show in several units at once, and rows that do not apply to the chosen shape are hidden.

The calculator assumes uniform material and sharp corners, and ignores fillets, rounded edges, holes you have not entered and welds. Real rolled sections have root radii and tapered flanges, so published tables for a product are more accurate than a simple I-beam rectangle model. The mass shapes assume a uniform density. Treat every result as an estimate, and follow the applicable standards, local building code and the manufacturer's data for real work.

For related tools, the rafter length calculator gives the span and length of a framing member whose bending stiffness depends on its section, the board foot calculator works out the volume of timber sections, and the percentage calculator finds the percent change between two section sizes.