Given a thin square lamina of a size a × a and mass M (see fig.)
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Sol. Denote by A the area of the square. By definition, in our case, we have :
I x = 
…..(i)
where σ is the mass per unit area. Therefore,
I x = 
....(ii)
Since M = σ a 2 . As the square is symmetric under rotation by 90º, we have
I y = I x …..(iii)
The moment of inertia around the z axis is
I z = 
= 
= 
+ 
= 
= 
…..(iv)
We have : I z = I x + I y = 
.....(v)
Hence, the perpendicular axes theorem is verified.
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