About the Moment of Inertia Calculator
The second moment of area, often called the moment of inertia of a section, tells you how much a cross-section resists bending. Pick a shape, type its dimensions in millimetres and this calculator returns Ix and Iy about the centroid in mm⁴ and cm⁴, the area, the section modulus on each axis and the radius of gyration. It covers rectangles, hollow boxes, solid circles, tubes, triangles and symmetric I-beams. The results feed straight into beam deflection and stress formulas. Everything is calculated in your browser, and the page checks your inputs, for example when a wall is too thick for the box.
Updated October 2026
How to use the Moment of Inertia Calculator
- 1Choose the cross-section shape.
- 2Type each dimension in millimetres.
- 3Read Ix and Iy in mm⁴ and cm⁴.
- 4Use the section modulus for bending stress and the radius of gyration for buckling.
- 5Copy the result into your beam calculation.
Why the shape matters more than the area
Two bars with the same amount of steel can have very different bending stiffness. A flat strip lying down bends easily, while the same strip standing on edge is much stiffer, because the second moment of area grows with the cube of the height. This is why floor joists are deep and narrow and why I-beams have most of their material in the flanges.
The numbers on this page are about the centroid. If you need the value about another axis, use the parallel axis theorem and add the area times the square of the distance between the axes.
Formulas used
- Rectangle: Ix = b h³ / 12.
- Solid circle: I = π D⁴ / 64.
- Tube: I = π (D⁴ − d⁴) / 64.
- Hollow box: outer rectangle minus the inner rectangle.
- Triangle about the centroid: Ix = b h³ / 36.
- Symmetric I-beam: outer rectangle minus the two side gaps beside the web.
Related tools
Feed the result to the Beam Deflection Calculator to see how a beam sags. The Area Calculator and the Circle Calculator give the area and perimeter of simple shapes, and the Triangle Calculator helps with triangular sections. Always have a qualified engineer check real structural designs.
Features
- Rectangle, hollow box, circle, tube, triangle and I-beam
- Ix and Iy about the centroid
- Area, section modulus and radius of gyration
- mm⁴ and cm⁴ side by side
- Checks for walls that are too thick
- No sign-up and nothing uploaded
Frequently asked questions
What is the moment of inertia of a rectangle?
About the horizontal centroidal axis it is b times h cubed divided by 12, where b is the width and h is the height.
Is this the same as mass moment of inertia?
No. The area moment of inertia describes a cross-section for bending. The mass moment of inertia describes how hard it is to spin an object. They have different units.
What is the section modulus?
It is the second moment of area divided by the distance from the centroid to the outer edge. Bending stress equals the moment divided by the section modulus.
What is the radius of gyration?
It is the square root of I divided by the area. Engineers use it in column buckling checks.
Which axis is x?
Ix is the bending resistance about the horizontal axis, so it uses the height h. Iy is about the vertical axis and uses the width.
How do I get an I-beam value for a standard section?
Standard steel sections have I listed in tables. This calculator is for a simplified symmetric I shape without fillets.
Why is a tube stronger in bending than a rod?
The material is placed farther from the centre, where it counts the most, since distance enters the formula to the fourth power for a circle.