Unique proof of the rotational inertia of a rectangular plate

Proof:

We're finding the rotational inertia of a rectangular plate with the axis passing through the CM, perpendicular to the plate. The mass of the plate is $M$, and the dimensions are $a,b$.

Lemma 1: Perpendicular axis theorem: $I_z = I_x + I_y$ for a planar object

Proof of lemma:

Since it's a planar object, the perpendicular distance from a point on the object to the $z$-axis is simply the distance from the point to the origin. This distance is $\sqrt{x^2+y^2}$, so we have$$I_z=m(x^2+y^2)=mx^2+my^2=I_y + I_x.$$ (By the way, no, $mx^2=I_y$ and $my^2 = I_x$ is not a typo!)

Lemma 2: The rotational inertia of an object around an axis does not change if the object stretches parallel to the axis.

Proof of lemma:

I'll start with an example: have you noticed that the rotational inertia of a cylinder and disk of the same mass and the same radius are the same? They are both $\frac{1}{2}MR^2$

This is exactly what I'm trying to prove: consider a disk, and you've just increased its length parallel to the axis. That won't change the rotational inertia, because for any point, this procedure does not change the perpendicular distance from the axis.

Generalizing this, note that increasing the length of any object parallel to the axis doesn't change it's rotational inertia, as long as we assume the mass of the initial and final object is the same. Another way to look at this is, the new object with increased length can be decomposed into the old object, stacked up several times. The rotational inertia of each stacked "old object" is the same, so summing all of these gives us just the same formula as of the old object, just with the mass changed. My point is, the formula for the rotational inertia won't change.

For the purpose of using Lemma 2, assume that the rectangular plate has $0$ thickness -- we will generalize this later. We want to find $I_z$, where our origin is located at the center of the plate. and using the theorem above, that's equal to $I_x+I_y$. Now, using Lemma 2, $I_x$ is just the rotational inertia of a rod with mass $M$ and length $a$. Similarly, $I_y$ is the rotational inertia of a rod with mass $M$ and length $b$. I have attached a diagram below to make this point clearer. 

Thus $I_x=\frac{1}{12}Ma^2$ and $I_y=\frac{1}{12}Mb^2$. So, we have$$I_z=I_x+I_y = \frac{1}{12}Ma^2 + \frac{1}{12}Mb^2 = \boxed{\frac{1}{12}M(a^2+b^2)},$$which is the formula we know.

Now, we can use Lemma 2 to generalize this to any rectangular plate of any thickness. $\square$