Moment of Inertia
Moment of inertia, also called the second moment of area, is the product of area and the square of its moment arm about a reference axis.

Moment of inertia about the x-axis:

$\displaystyle I_x = \int y^2 \, dA$

Moment of inertia about the y-axis:

$\displaystyle I_y = \int x^2 \, dA$

Polar Moment of Inertia:
Polar moment of inertia is the moment of inertia about about the z-axis.

$J = I_x + I_y$

$\displaystyle J = \int r^2 \, dA$

$k = \sqrt{\dfrac{I}{A}}$

$k_x = \sqrt{\dfrac{I_x}{A}}$

$k_y = \sqrt{\dfrac{I_y}{A}}$

$k_z = \sqrt{\dfrac{J}{A}}$

Transfer Formula for Moment of Inertia

$I = \bar{I} + Ad^2$

Where
$x'$ = centroidal axis
$x$ = any axis parallel to the centroidal axis
$I$ = moment of inertia about the x-axis
$\bar{I}$ = centroidal moment of inertia
$A$ = area of the section
$d$ = distance between x and x’

In the same manner, the transfer formula for polar moment of inertia and the radii of gyration are respectively

$J = \bar{J} + Ad^2$

$k^2 = {\bar{k}}^2 + d^2$

Product of Inertia

$\displaystyle I_{xy} = \int xy \, dA$

Moment of Inertia of Common Shapes

 Shape Moment of Inertia Radius of Gyration Rectangle $\bar{I}_x = \dfrac{bh^3}{12}$ $I_x = \dfrac{bh^3}{3}$ $\bar{k}_x = \dfrac{h}{\sqrt{12}}$ $k_x = \dfrac{h}{\sqrt{3}}$ Triangle $\bar{I}_x = \dfrac{bh^3}{36}$ $I_x = \dfrac{bh^3}{12}$ $\bar{k}_x = \dfrac{h}{\sqrt{18}}$ $k_x = \dfrac{h}{\sqrt{6}}$ Circle $\bar{I}_x = \dfrac{\pi r^4}{4}$ $\bar{J} = \dfrac{\pi r^4}{2}$ $\bar{k}_x = \dfrac{r}{2}$ $\bar{k}_z = \dfrac{r}{\sqrt{2}}$ Semicircle $I_x = \bar{I}_y = \dfrac{\pi r^4}{8}$ $\bar{I}_x = 0.11r^4$ $k_x = \bar{k}_y = \dfrac{r}{2}$ $\bar{k}_x = 0.264r$ Quarter circle $I_x = I_y = \dfrac{\pi r^4}{16}$ $\bar{I}_x = \bar{I}_y = 0.055r^4$ $k_x = k_y = \dfrac{r}{2}$ $\bar{k}_x = \bar{k}_y = 0.264r$ Ellipse $\bar{I}_x = \dfrac{\pi ab^3}{4}$ $\bar{I}_y = \dfrac{\pi a^3b}{4}$ $\bar{k}_x = \dfrac{b}{2}$ $\bar{k}_y = \dfrac{a}{2}$

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