Moment of inertia

Ok let's make a coordinate system and put the origin in centar of our rectangle, let z axis be our axis, x parallel with b and y parallel with c. Now take one small piece A with coordinates $x, y$ and mass $m_A$ his moment inertia will be $I_A=m_A r^2=m_A (x^2+y^2)=I_Ax+I_Ay$

So now we see that $I=I_x+I_y$ .

To calculate $I_x$ and $I_y$ we will devide our rectangle into two groups of rods. First parallel to $a$ with lenght $a$ and second parallel with $b$ with lenght $b$ . And now you get

$I_x=\frac{ma^2}{12}$ and $I_y=\frac{mb^2}{12}$

And finally

$I=\frac{m(a^2+b^2)}{12}=25$