Parallel Axis Theorem

$$I_x = I_{x_1} + A\bar{y}^2$$ $$I_y = I_{y_1} + A\bar{x}^2$$ $$I_{xy} = I_{x_1 y_1} + A\bar{x}\bar{y}$$

Here

• On LHS, the moments of area are about some $x$, $y$ axes.
• On RHS, the moments of area are about centroidal axes $x_1$, $y_1$ parallel to x, y.
• $\bar{x}$ is the distance between $x$ and $x_2$ axes.
• $\bar{y}$ is the distance between $y$ and $y_1$ axes.

Note

$I_x$ is at a minimum when the axis is through the centroid. Same for $I_y$.