Moment of Inertia & Centroid Calculator

A rectangle. A 100 mm wide, 200 mm deep section: A = 100 × 200 = 20,000 mm², the centroid is at mid-depth, and Ix = bh³/12 = 100 × 200³ ÷ 12 = 66,666,667 mm⁴ (6,666.7 cm⁴). The extreme fibre is c = 100 mm away, so S = Ix ÷ c = 666,667 mm³.

A T-section, part by part. Flange 150 × 12 mm (A = 1,800 mm², centre 144 mm up) on a stem 10 × 138 mm (A = 1,380 mm², centre 69 mm up). The centroid is ȳ = (1,800 × 144 + 1,380 × 69) ÷ 3,180 = 111.45 mm from the bottom. About that axis the flange gives 150 × 12³ ÷ 12 + 1,800 × 32.55² = 1,928,373 mm⁴ and the stem 10 × 138³ ÷ 12 + 1,380 × 42.45² = 4,677,155 mm⁴, so Ix = 6,605,528 mm⁴. The bottom of the stem is 111.45 mm from the axis and the top 38.55 mm, so the least section modulus is 6,605,528 ÷ 111.45 = 59,267 mm³.

The common mistake: adding each part’s own I and forgetting A·d². For a 200 × 100 I-beam (8.5 mm flanges, 5.6 mm web) the two flanges have only 5,118 mm⁴ each about their own centres, and the web 2,859,961 mm⁴. Adding those gives 2,870,196 mm⁴. The right answer moves each flange 95.75 mm to the neutral axis, adding 850 × 95.75² = 7,792,853 mm⁴ each, for Ix = 18,455,902 mm⁴, 6.4 times more. That distance term is the whole reason I-beams work.

Galileo asked how strong a beam is in Two New Sciences (1638). He saw that a beam’s strength grows with its depth faster than with its width, but he placed the turning point at the bottom edge of the section, so his numbers were too high. Edme Mariotte (1686) and Antoine Parent (1713) moved the axis inside the section, and Claude-Louis Navier settled it in his 1826 lectures: the neutral axis of an elastic beam passes through the centroid of the section, and stiffness depends on the integral of y² over the area, the quantity this calculator computes.

The rotating-body version is older in spirit. Christiaan Huygens worked out the swing of a compound pendulum in Horologium Oscillatorium (1673) using sums of mass times distance squared, and Leonhard Euler gave the quantity its name, the moment of inertia, in his 1765 book on the motion of rigid bodies. The rule for shifting it to a parallel axis is often called the Huygens–Steiner theorem, after Huygens and the Swiss geometer Jakob Steiner.

The rolled-iron I-beam that puts the idea into practice appeared in the mid-19th century: most of the metal sits in the flanges, far from the axis, where the A·d² term multiplies its value.

This calculator builds each section from rectangles and circles, adds holes as negative areas, and finds the centroid and the second moments about it with the parallel-axis theorem. Show Work lists every part with its area, its centre, its own moment and the A·d² term, so you can check it against a hand calculation. It also gives the section modulus to the top and bottom fibres separately for shapes that are not symmetric, the radii of gyration, the polar moment, and for angles the product of inertia and the principal axes. The drawing is to scale, with the centroid and both neutral axes marked.

Rolled steel sections have rounded root fillets and toes that a sharp-cornered model leaves out, so a catalogue figure for the same nominal size can differ by a few percent; use the catalogue value for design. The Solid body mode gives the mass moment of inertia of ten common shapes, the radius of gyration and the energy stored at 1,000 rpm. Everything runs in your browser; nothing you enter is sent anywhere.

Related tools: Beam Calculator, Torque Calculator, and Density, Mass & Volume Calculator.