Again, a extract from the handbook gives all of these properties for

various shapes.

But mostly, we'll be looking here at the triangle, rectangle, and the circle.

And the moment of inertia of a rectangle about its centroid

is the width times the height cubed over 12 and

the moment of inertia of a circle about its centroid

is pi times the radius to the 4th power divided by 4.

And of course by symmetry,

the centroids of both of those shapes are exactly in the middle.

So those are probably the shapes that we'll be mostly concerned with.

Let me show that by an example.

We have a rectangle which is four meters tall by one meter wide, and

the first question, the moment of inertia about an axis parallel to the axis through

the centroid is most nearly which of these alternatives.

So, here's a centroid.

And, we want to get the moment of inertia

about a horizontal axis through that centroid.