Area Moment of Inertia Calculator

Calculator

Shape

Length unit

Rotation angle, degrees

Offset to parallel x axis, dy

Offset to parallel y axis, dx

Solid rectangle dimensions

Width b

Height h

Composite rectangle parts

Use x and y as the lower left corner of each rectangle. Choose void for holes or cutouts.

Part	Type	Width	Height	x	y
1
2
3
4

Formula used

The calculator uses standard centroidal area formulas for each selected shape. For shifted axes, it applies the parallel axis theorem: I = Ic + A d². For composite rectangles, it sums signed areas, finds the centroid, and shifts each part to the final centroid. For rotation, it uses transformed inertia equations with Ix, Iy, Ixy, and the selected angle.

How to use this calculator

Select the shape that matches your section.
Choose one length unit and keep every dimension in that unit.
Enter the needed dimensions for the selected shape.
Add offset distances if you need a parallel reference axis.
Enter a rotation angle if you need transformed axes.
Press Calculate, or download the same result as CSV or PDF.

Example data table

Shape	Key dimensions	Area	Ix about centroid	Iy about centroid
Rectangle	b = 100 mm, h = 200 mm	20,000 mm²	66,666,666.67 mm⁴	16,666,666.67 mm⁴
Circle	d = 100 mm	7,853.98 mm²	4,908,738.52 mm⁴	4,908,738.52 mm⁴
Hollow rectangle	b = 160 mm, h = 240 mm, bi = 100 mm, hi = 160 mm	22,400 mm²	147,626,666.67 mm⁴	66,986,666.67 mm⁴

Why area moment matters

Area moment of inertia describes how a cross section spreads area around an axis. A larger value means the shape resists bending better about that axis. It is not mass inertia. It only depends on geometry. Beam checks, shaft supports, machine frames, brackets, and rails all use this property. The same area can perform very differently when material is moved farther from the neutral axis.

Advanced section options

This calculator covers solid, hollow, and built up shapes. You can select rectangles, tubes, circles, triangles, I sections, T sections, channels, or a composite rectangle model. The composite option lets you combine solid rectangles and void rectangles. This is useful for boxed frames, cut plates, welded layouts, and quick design studies. Enter every dimension in one length unit. The result uses that unit raised to the fourth power.

Centroid and axis control

Many design tasks need inertia about the centroidal axis. Others need inertia about an offset reference line. The calculator reports both. It first finds the centroid. It then applies the parallel axis theorem when you enter horizontal or vertical offsets. Positive offsets are distances from the centroid to your chosen reference axis. This helps when a section is attached to another member or checked from a datum.

Rotated and principal axes

Sections may not always bend around their strong or weak axis. A rotated beam, angled plate, or unsymmetric group can need transformed inertia values. The angle field rotates the centroidal axes and reports transformed moments and product inertia. Principal results show the maximum and minimum centroidal inertia values. These values are helpful when checking the most efficient orientation.

Practical use

Use dimensions from a drawing, not rounded catalog names. Check that hollow dimensions fit inside outer dimensions. For thin parts, small changes in height can create large changes in inertia because height is cubed in many formulas. Compare Ix and Iy before choosing an orientation. Download the CSV for spreadsheets. Download the PDF when you need a simple design record. For final work, always compare results with project standards and manufacturer data or codes. This tool supports early sizing, classroom study, and review notes. It does not replace licensed engineering judgment for safety critical designs.

FAQs

What is area moment of inertia?

It is a geometric property that shows how area is distributed around an axis. It helps estimate bending stiffness for beams and plates.

Is this the same as mass moment of inertia?

No. Area moment uses section geometry only. Mass moment also uses mass distribution and is used for rotational dynamics.

Which unit should I use?

Use any listed length unit, but keep every input in that same unit. The inertia output is that unit to the fourth power.

What does Ix mean?

Ix is the second moment of area about the x axis. It is commonly used for bending caused by loads that curve a beam vertically.

What does the parallel axis option do?

It shifts centroidal inertia to another parallel axis. The calculator adds A times the square of the offset distance.

When should I use composite rectangles?

Use that option for built sections, plate groups, or cutouts that can be modeled as added and removed rectangles.

Why are principal values useful?

Principal values show the maximum and minimum centroidal inertia for rotated axes. They help find strong and weak bending directions.

Can I use results for final design?

Use results for study and preliminary checks. Final structural work should follow project codes, tolerances, and qualified engineering review.