Counting sides or edges of polygon using open source GIS?

With Python and Fiona, Polygons and MutiPolygons (multi-parts) are different geometries:

1) multi-parts geometries

 import fiona
 shape = fiona.open("polygons.shp")
 # shapefile schema
 print c.schema
 {'geometry': 'Polygon', 'properties': OrderedDict([(u'id', 'int:10')])}
 # first feature
 first = shape.next()
 print first (GeoJSON format)
 {'geometry': {'type': 'Polygon', 'coordinates': [[(244697.45179524383, 1000369.2307574936), (244827.15493968062, 1000373.0455558595), (244933.96929392271, 1000353.9715640305), (244933.96929392271, 1000353.9715640305), (244930.15449555693, 1000147.9724522779), (244697.45179524383, 1000159.4168473752), (244697.45179524383, 1000369.2307574936)]]}, 'type': 'Feature', 'id': '0', 'properties': OrderedDict([(u'id', 1)])}


 multi = fiona.open("multipolygons.shp")
 print c.schema
 {'geometry': 'Polygon', 'properties': OrderedDict([(u'id', 'int:10')])}
 # first feature
 first = shape.next()
 print first
 {'geometry': {'type': 'MultiPolygon', 'coordinates': [[[(244697.45179524383, 1000369.2307574936), (244827.15493968062, 1000373.0455558595), (244933.96929392271, 1000353.9715640305), (244933.96929392271, 1000353.9715640305), (244930.15449555693, 1000147.9724522779), (244697.45179524383, 1000159.4168473752), (244697.45179524383, 1000369.2307574936)]], [[(246082.22360202507, 1000453.1563215409), (246139.44557751188, 1000460.7859182726), (246189.03795626713, 1000403.5639427857), (246189.03795626713, 1000403.5639427857), (246086.03840039085, 1000132.7132588148), (245990.66844124615, 1000205.1944277647), (246082.22360202507, 1000453.1563215409)]]]}, 'type': 'Feature', 'id': '0', 'properties': OrderedDict([(u'id', 1)])}

2) With simple Polygons, the coordinates are

coord = first['geometry']['coordinates']
[[(244697.45179524383, 1000369.2307574936), (244827.15493968062, 1000373.0455558595), (244933.96929392271, 1000353.9715640305), (244933.96929392271, 1000353.9715640305), (244930.15449555693, 1000147.9724522779), (244697.45179524383, 1000159.4168473752), (244697.45179524383, 1000369.2307574936)]
print len(coord)
1 # -> one polygon

And the equivalent LinearRing coordinates:

linearR = coor[0] # or coord = first['geometry']['coordinates'][0]
print linearR
[(244697.45179524383, 1000369.2307574936), (244827.15493968062, 1000373.0455558595), (244933.96929392271, 1000353.9715640305), (244933.96929392271, 1000353.9715640305), (244930.15449555693, 1000147.9724522779), (244697.45179524383, 1000159.4168473752), (244697.45179524383, 1000369.2307574936)]

So

nb_vertices = len(lineaR) #(x,y) points
print nb_vertices
7 
nb_edges = nb_vertices -1
print nb_edges
6

3) With MultiPolygons, use a for loop

4) If the Polygons have holes, use Shapely with Fiona

holes = fiona.open("poly_holes.shp")
from shapely.geometry import shape
# First feature
first = holes.next()
print first['geometry']['coordinates']
# conversion to shapely geometry
shape = first['geometry']
[[(1.0, 1.0), (1.0, 7.0), (7.0, 7.0), (7.0, 1.0), (1.0, 1.0)], [(2.0, 3.0), (4.0, 3.0), (4.0, 5.0), (2.0, 5.0), (2.0, 3.0)], [(5.0, 5.0), (6.0, 5.0), (6.0, 6.0), (5.0, 6.0), (5.0, 5.0)]]
# exterior coordinates
print list(shape.exterior.coords) # = LinearRing
[(1.0, 1.0), (1.0, 7.0), (7.0, 7.0), (7.0, 1.0), (1.0, 1.0)]
edges_ext= len(list(shape.exterior.coords))
print edges_ext
5

Same with interior coordinates (look at the Shapely manual)

In QGIS if your polygons do not have any holes or multi-parts:

l = iface.activeLayer()
for f in l.getFeatures():
    print f['NAME']
    print 'no. edges: %d' %(len(f.geometry().asPolygon()[0])-1)

replace 'NAME' with some identifier in your layer attribute table.

Concerning writing to the attribute table check the instructions in the PyQGIS Cookbook - Modifying Vector Layers.