Moments of inertia of a rectangular plate

[Karol]

Homework Statement

Show that the moment of inertia of a rectangular plate round it's diagonal, line B-B is equal to the one round a line parallel to one of the edges and passing through the center, line C-C

Homework Equations

Moment of inertia: ##I=mr^2##

The Attempt at a Solution

For the first case, round the diagonal: the MOI of a straight angle and isosceles triangle round one of the 2 identical edges is ##I=\frac{1}{6}ma^2##, and if i divide the plate into 4 such rectangles and calculate MOI it equals the one round line C-C.
But is there an easier way?

 

[Orodruin]

Depending on your background knowledge, the symmetry of the square will tell you that they have to be equal.

 

[Karol]

Depending on your background knowledge, the symmetry of the square will tell you that they have to be equal.

I have little background knowledge, please hint the subject that i have to study, is it principal axes? as far as i (think) i know i can't know anything about the MOI itself around principal axis, especially if they aren't orthogonal, but i am not sure.
Please guide me to a book or publication that explains this topic, but, if possible, without the complexity of determinants, tensors etc.

 

[Orodruin]

You can do it with principal axes, but all that is required is noting that an axis through the corners is a linear combination of the axes through the center and the center of the sides, which must have the same moment of inertia due to symmetry.

 

[Karol]

all that is required is noting that an axis through the corners is a linear combination of the axes through the center and the center of the sides, which must have the same moment of inertia due to symmetry.

So i just have to know the formula for modifying MOI with the changing of the angle of the axis?

 

[Orodruin]

If you want to call it a formula, fine. The point is that any axis in the plane of the square has the same moment of inertia because the axes spanning the plane do.