  In this paper, we first consider a flat plate (called a lamina) with uniform
density $\rho$ that occupies a region $\mathfrak R$ of the plane. We show that
the location of the center of mass, also known as the centroid, of the region
equals the expected vector of a bivariate continuous random variable with a
uniform probability distribution taking values on the region $\mathfrak R$.
Using this property, we prove that the Voronoi regions of an optimal set of
two-means with respect to the uniform distribution defined on a disc partition
the disc into two regions bounded by the semicircles. Besides, we show that if
an isosceles triangle is partitioned into an isosceles triangle and an
isosceles trapezoid in the Golden ratio, then their centers of mass form a
centroidal Voronoi tessellation of the triangle. In addition, using the
properties of center of mass we determine the optimal sets of two-means and the
corresponding quantization error for a uniform distribution defined on a region
with uniform density bounded by a rhombus. Further, we determine the optimal
sets of $n$-means, and the $n$th quantization errors for two different discrete
uniform distributions for some positive integers $n\leq \text{card(supp}(P))$.
