Online structural engineering calculators

Calculate the moment of inertia of the sections most used in structural engineering: square, rectangular, circular, hollow square, ring-shaped and hollow rectangular. Pick a tool from the menu on the left or from the cards below; formulas, notes and frequently asked questions are further down.

Featured tools

CalculatorRectangular sectionJ = b · h³ / 12, about the centroidal axis. CalculatorCircular sectionJ = π · d⁴ / 64, from the diameter. CalculatorSquare sectionJ = a⁴ / 12, from the side.

Moment of inertia

Six calculators for the moment of inertia of solid and hollow sections. Choose the section: the tool opens at the top of this page.

CalculatorSquare sectionJ = a⁴ / 12. Solid section with side a. CalculatorRectangular sectionJ = b · h³ / 12. Solid section with base b and height h. CalculatorCircular sectionJ = π · d⁴ / 64. Solid section with diameter d. CalculatorHollow square sectionJ = (A⁴ − a⁴) / 12. Square with a centered square hole. CalculatorRing sectionJ = π · (D⁴ − d⁴) / 64. Hollow circular tube. CalculatorHollow rectangular sectionJ = (B · H³ − b · h³) / 12. Rectangle with a centered hole.

Main formulas

The moment of inertia J of a section measures its resistance to bending: the larger it is, the smaller the deformation under the same load. The formulas below hold about the centroidal axis and, for hollow sections, with a centered hole.

Section	Moment of inertia	Symbols
Square	J = a⁴ / 12	a side
Rectangular	J = b · h³ / 12	b base, h height (axis parallel to the base)
Circular	J = π · d⁴ / 64	d diameter
Hollow square	J = (A⁴ − a⁴) / 12	A outer side, a inner side
Ring	J = π · (D⁴ − d⁴) / 64	D outer diameter, d inner diameter
Hollow rectangular	J = (B · H³ − b · h³) / 12	B, H outer dimensions; b, h inner dimensions

Moment of inertia about the centroidal axis of the most common sections.

Practical notes

The moment of inertia is expressed in units of length to the fourth power, for example cm⁴ or mm⁴, with 1 cm⁴ = 10,000 mm⁴. It depends on the cube of the height: doubling the height of a rectangle makes J eight times larger. It appears in the deflection formula: for a simply supported beam with a uniformly distributed load q and span L, δ = 5 · q · L⁴ / (384 · E · J). The elastic section modulus is obtained as W = J / y, where y is the distance from the axis to the farthest point of the section.

Frequently asked questions

What is the moment of inertia of a section?

It is a geometric quantity that measures the resistance of the section to bending about an axis: J = ∫ y² dA. Under the same load, a section with a larger J deflects less. It is measured in cm⁴ or mm⁴.

How do you calculate the moment of inertia of a rectangle?

With J = b · h³ / 12 about the centroidal axis parallel to the base. Example: b = 10 cm and h = 20 cm give J = 10 · 8,000 / 12 ≈ 6,666.67 cm⁴.

What is the moment of inertia of a circular section?

J = π · d⁴ / 64, where d is the diameter. Example: for d = 10 cm, J = π · 10,000 / 64 ≈ 490.87 cm⁴.

What is the difference between a solid and a hollow section?

A ring with D = 10 cm and d = 8 cm has J = π · (10,000 − 4,096) / 64 ≈ 289.8 cm⁴. Compared with the solid 10 cm circle, it uses about 36% of the area and keeps about 59% of the moment of inertia.