Editing Pappus's centroid theorem (section)

==Generalizations==

The theorems can be generalized for arbitrary curves and shapes, under appropriate conditions. 

Goodman & Goodman<ref name=generalizations>{{cite journal | last1=Goodman | first1=A. W. | last2=Goodman | first2=G. | title = Generalizations of the Theorems of Pappus | journal=The American Mathematical Monthly |volume=76 |issue=4 |pages=355–366 | jstor=2316426 | year = 1969 | doi = 10.1080/00029890.1969.12000217}}</ref> generalize the second theorem as follows. If the figure {{math|''F''}} moves through space so that it remains [[perpendicular]] to the curve {{math|''L''}} traced by the centroid of {{math|''F''}}, then it sweeps out a solid of volume {{math|1=''V'' = ''Ad''}}, where {{math|''A''}} is the area of {{math|''F''}} and {{math|''d''}} is the length of {{math|''L''}}. (This assumes the solid does not intersect itself.) In particular, {{math|''F''}} may rotate about its centroid during the motion. 

However, the corresponding generalization of the first theorem is only true if the curve {{math|''L''}} traced by the centroid lies in a plane perpendicular to the plane of {{math|''C''}}.