Area moment of inertia formula
The area moment of inertia describes how a cross-section resists bending, based on how its area is distributed about the neutral axis. Area placed farther from the axis counts much more, which is why beam shapes look the way they do.
Variables
| I | Area moment of inertia (rectangle) | in⁴ or m⁴ |
| b | Width | in or m |
| h | Height (depth) | in or m |
Rearranged
Section modulus: S = I / c
Worked example
A solid rectangle is 50 mm wide and 100 mm deep.
Use I = b h³ / 12: 50 × 100³ / 12 = 4,166,667 mm⁴.
Area moment of inertia governs both bending stress and deflection: a higher I means a stiffer, stronger beam. Because depth is cubed in the rectangle formula, moving material away from the neutral axis is enormously effective, which is the reason for tall beams and I-shaped sections.
Need section modulus or deflection next?
See the Section Modulus Formula and the Beam Deflection Formula.
What area moment of inertia means
It is a purely geometric property: each bit of area is weighted by the square of its distance from the neutral axis and summed. Area near the axis barely helps; area far from it helps a great deal. This is distinct from the mass moment of inertia used in rotation, and from the polar moment used in torsion.
Common shapes
For a solid rectangle bending about its horizontal centroid, I equals width times depth cubed over twelve. For a solid circle, I equals pi times diameter to the fourth over sixty-four. Hollow and built-up shapes are found by subtracting the hole or adding the parts, which is how I-beams and tubes get their high efficiency.
FAQ
What is the area moment of inertia formula?
For a rectangle, I equals width times depth cubed over twelve. For a circle, I equals pi times diameter to the fourth over sixty-four.
What does area moment of inertia tell you?
How well a cross-section resists bending and deflection. A higher value means a stiffer, stronger beam for the same material.
Is area moment of inertia the same as mass moment of inertia?
No. Area moment of inertia is about a cross-section resisting bending; mass moment of inertia is about a body resisting angular acceleration.