question on finding moment of inertia about the center of mass

So I have a question. I (moment of inertia) is basically mr^2 right? And r is supposed to be the distance from the axis of rotation. When the axis of rotation is directly through the center of mass, how is there Icm (moment of inertia about the center of mass). It’s confusing to me, because so many problems demand using the Icm when the axis is through the center of mass. Wouldn’t Icm be zero? But then they expect you to use the radius of the cylinder. The I of a solid cylinder is given as 1/2(Mr^2). Someone please help me. I think it might just be my lack of understanding.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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