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For this question we are working in 2-D space. Let's say that we have an arbitrary body. I know the inertia with respect to a perpendicular axis (to the body, that is) that passes through a point (A). However, I'd like to know the inertia with respect to a parallel axis (B) a distance $d$ away so I use the parallel axis theorem:

$$ I_B = I_\mathrm{A} + md^2 $$

I show the diagram of the 2-D body and my value $I_B$ (not $I_A$) to someone else. They, however, would like to know what the inertia is with respect to the axis at A. They use the parallel axis theorem - plug n' chug - as well.

$$ I_A = I_\mathrm{B} + md^2 $$

For $d \in \mathbb{R}, \quad d^2$ is bound to be > 0! So their $I_A$ is not going to equal the initial $I_A$. Obviously there has been some confusion of ideas on my part. But I don't know where I went wrong.

Where did I go wrong?

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The parallel axis theorem $I_B = I_A + md^2$ only applies when $A$ is the center of mass.

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