Figure: The centers of mass of symmetric bodies is at the geometric centers

Caption: A diagram illustrating the centers of mass of 2-dimensional objects of uniform density with high symmetry.

Recall, a center of mass is the mass-weighted average position of an object.

You can find these centers of mass of the objects in the diagram by inspection. They are just the geometric centers or centers of symmetry.

Further explication:

1. The centers of mass of the displayed objects must be the points of highest symmetry (i.e., the geometric centers) because where else could they be if the symmetry is NOT to be broken.

   The points of highest symmetry are the points about which each object can be rotated into itself by any angle which is multiple of a unit angle specific to the object: e.g., infinitesimal for the circle, 90° for the square, 72° for the regular pentagon and pentagram, etc.

2. Many 3-dimensional objects with simple shapes and uniform density also have centers of symmetry which are also centers of mass that can be found by inspection: e.g.,
   cube, cylinder, ellipsoid, parellelepiped, Platonic solid, regular tetrahedron, sphere, spheriod (AKA ellipsoid of revolution), torus, etc.

3. More details about 2-dimensional objects of high symmetry and uniform density are given in File: Mechanics file: center_of_mass_2d_4.html.