Mass and geometry properties (my scratchpad)

Volume, center of mass and matrix of inertia

For all solids you can get:

1. volume - add Shape.Volume attribute to object,
2. center of mass - add Shape.CenterOfMass attribute,
3. matrix of inertia - add Shape.MatrixOfInertia attribute.

Example:
Cone (height = 40 mm, radius = 30 mm) named "Revolution"

>>> App.ActiveDocument.getObject("Revolution").Shape.MatrixOfInertia
Matrix ((7.35133e+06,-6.48484e-11,2.72874e-09,0),(-6.48484e-11,7.35133e+06,4.02739e-10,0),(2.72874e-09,4.02739e-10,1.01788e+07,0),(0,0,0,1))
>>> App.ActiveDocument.getObject("Revolution").Shape.CenterOfMass
Vector (6.11835e-15, 6.99401e-16, 30)
>>> App.ActiveDocument.getObject("Revolution").Shape.Volume
37699.11184307751

MatrixOfInertia attribute output needs some explanation:
http://free-cad.svn.sourceforge.net/viewvc/free-cad/trunk/src/Mod/Part/App/TopoShapeSolidPy.xml?view=markup&pathrev=5337

Returns the matrix of inertia. It is a symmetrical matrix.
The coefficients of the matrix are the quadratic moments of inertia.

 | Ixx Ixy Ixz 0 |
 | Ixy Iyy Iyz 0 |
 | Ixz Iyz Izz 0 |
 | 0 0 0 1 |


The moments of inertia are denoted by Ixx, Iyy, Izz.
The products of inertia are denoted by Ixy, Ixz, Iyz.
The matrix of inertia is returned in the central coordinate system (G, Gx, Gy, Gz) where G is the centre of mass of the system and Gx, Gy, Gz the directions parallel to the X(1,0,0) Y(0,1,0) Z(0,0,1) directions of the absolute cartesian coordinate system.

So, we have 3 moments of inertia:
Ixx=7.35133e+06
Iyy=7.35133e+06
Izz=1.01788e+07

Further explanation - FreeCAD forum.

Area of face

You can measure area of face, simply add Area attribute. If you need area of a sketch, convert one to a face:
>>> face = Part.Face(App.ActiveDocument.getObject("Sketch").Shape)
>>> face.Area
1469.9597436001611

Update:
If you have a solid (eg. a Pad), select face or faces on the solid object, and use this script:

area = 0.0
for o in Gui.Selection.getSelectionEx() :
        for s in o.SubObjects:
                area = s.Area
                print "Area of selected face:" ,area