Moment of Inertia Formulas for Common Shapes

The moment of inertia (second moment of area) measures how a section resists bending. Each formula below is about the centroidal axis parallel to the width. Lengths can be in any unit; the result is in that unit to the fourth power.

Moment of inertia formulas
Shape	Formula	Symbols
Rectangle	$I = b \cdot h^{3} /12$	b = width, h = depth
Hollow rectangle	$I = (B \cdot H^{3} - b \cdot h^{3}) /12$	B, H outside; b, h inside
Circle	$I = π \cdot D^{4} /64$	D = diameter (= π·r⁴/4)
Hollow circle (pipe)	$I = π \cdot (D^{4} - d^{4}) /64$	D outside, d inside diameter
Triangle	$I = b \cdot h^{3} /36$	about the centroid, parallel to the base
I-beam (symmetric)	$I = (b_{f} \cdot h^{3} - (b_{f} - t_{w}) \cdot h_{w}^{3}) /12$	h_w = h − 2t_f

Parallel-axis theorem

To move a moment of inertia from a part's own centroid to another axis, add its area times the distance squared. This is how T-sections, I-beams and other built-up shapes are added together.

I = I_{c} + A \cdot d^{2}

Worked examples

Calculate it for your section

Rectangle, hollow rectangle, circle, T- and I-sections, in mm or inches.

Open the Moment of Inertia Calculator →

Moment of Inertia Formulas

Parallel-axis theorem

Worked examples

More free tools