Moment of Inertia Notes

Last reviewed 16 Sept 2026 · 5 min read

Second moment of area

The moment of inertia of an area (second moment of area) about an axis is the sum of each elemental area multiplied by the square of its distance from the axis:

I_{xx} = \int y^{2} d A, I_{y y} = \int x^{2} d A

Units: mm⁴ or m⁴. It measures how the area is spread away from the axis, and so governs a beam's resistance to bending ( $σ = M y / I$ , deflection $\propto 1/ E I$ ) and a column's resistance to buckling.

Mass moment of inertia $I = \int r^{2} d m$ (kg·m²) measures resistance to angular acceleration in dynamics — a different quantity with the same mathematical form.

Radius of gyration

k = \frac{I}{A} \Leftrightarrow I = A k^{2}

It is the distance from the axis at which the whole area could be concentrated to give the same moment of inertia. Columns buckle about the axis of least radius of gyration.

Theorems

FormulaParallel axis theorem

I_{A B} = I_{G} + A h^{2}

$I_{G}$ = moment of inertia about a centroidal axis; $h$ = distance to a parallel axis AB. The centroidal moment of inertia is the minimum among all parallel axes.

FormulaPerpendicular axis theorem (plane areas)

I_{z z} = I_{xx} + I_{y y} = J

The moment of inertia about an axis perpendicular to the plane (through the point where $x$ and $y$ meet) equals the sum of those about the two in-plane axes. This is the polar moment of inertia $J$ .

Standard values

Section	About	Moment of inertia
Rectangle $b \times d$	Centroidal axis parallel to $b$	$\frac{b d ^{3}}{12}$
Rectangle $b \times d$	Base (side $b$ )	$\frac{b d ^{3}}{3}$
Hollow rectangle	Centroidal	$\frac{B D ^{3} - b d ^{3}}{12}$
Triangle, base $b$ , height $h$	Centroidal axis parallel to base	$\frac{b h ^{3}}{36}$
Triangle	Base	$\frac{b h ^{3}}{12}$
Triangle	Axis through apex parallel to base	$\frac{b h ^{3}}{4}$
Circle, diameter $d$	Any diameter	$\frac{π d ^{4}}{64}$
Circle	Polar (centre)	$\frac{π d ^{4}}{32}$
Hollow circle $D$ , $d$	Diameter	$\frac{π ( D ^{4} - d ^{4} )}{64}$
Semicircle, radius $r$	Diameter (base)	$\frac{π r ^{4}}{8}$
Semicircle	Centroidal axis parallel to base	$\approx 0.11 r^{4}$ (exactly $(\frac{π}{8} - \frac{8}{9 π}) r^{4}$ )
Quadrant	Either straight edge	$\frac{π r ^{4}}{16}$
Quadrant	Centroidal axis parallel to an edge	$\approx 0.055 r^{4}$
Square of side $a$	Diagonal	$\frac{a ^{4}}{12}$ (same as about a centroidal axis parallel to a side)
Ellipse, semi-axes $a$ (along $x$ ), $b$	$x$ -axis	$\frac{π a b ^{3}}{4}$

RememberRadius of gyration shortcuts

Rectangle about centroidal axis: $k = d / 12$ . Solid circle: $k = d /4$ . Hollow circle: $k = D^{2} + d^{2} /4$ .

Composite sections

Divide the section into simple parts; locate the overall centroid.
Find each part's centroidal $I$ .
Transfer to the overall centroidal axis with $A h^{2}$ .
Add (subtract for holes).

Product of inertia and principal axes

I_{x y} = \int x y d A, I_{x y} = I_{x^{'} y^{'}} + A \overset{x}{ˉ} \overset{y}{ˉ} (parallel axis form)

$I_{x y} = 0$ if either axis is an axis of symmetry.
Principal axes: $tan 2 θ = \frac{2 I _{x y}}{I _{y y} - I _{xx}}$ ; principal values $\frac{I _{xx} + I _{y y}}{2} \pm (\frac{I _{xx} - I _{y y}}{2})^{2} + I_{x y}^{2}$ .

(See Unsymmetrical Bending & Shear Centre for applications.)

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Second moment of area

Radius of gyration

Theorems

Standard values

Composite sections

Product of inertia and principal axes

This chapter is in the syllabus of