  We present a new method to determine the probability distribution of the 3-D
shapes of galaxy clusters from the 2-D images using stereology. In contrast to
the conventional approach of combining different data sets (such as X-rays,
Sunyaev-Zeldovich effect and lensing) to fit a 3-D model of a galaxy cluster
for each cluster, our method requires only a single data set, such as X-ray
observations or Sunyaev-Zeldovich effect observations, consisting of
sufficiently large number of clusters. Instead of reconstructing the 3-D shape
of an individual object, we recover the probability distribution function (PDF)
of the 3-D shapes of the observed galaxy clusters. The shape PDF is the
relevant statistical quantity which can be compared with the theory and used to
test the cosmological models. We apply this method to publicly available
\emph{Chandra} X-ray data of 89 well resolved galaxy clusters. Assuming
ellipsoidal shapes, we find that our sample of galaxy clusters is a mixture of
prolate and oblate shapes, with a preference for oblateness with the most
probable ratio of principle axes 1.4 : 1.3 : 1. The ellipsoidal assumption is
not essential to our approach and our method is directly applicable to
non-ellipsoidal shapes. Our method is insensitive to the radial density and
temperature profiles of the cluster. Our method is sensitive to the changes in
shape of the X-ray emitting gas from inner to outer regions and we find
evidence for variation in the 3-D shape of the X-ray emitting gas with distance
from the centre.

Cosmology and Nongalactic Astrophysics;High Energy Astrophysical Phenomena