  The concept of the cyclic averages are introduced for a regular polygon $P_n$
and a Platonic solid $T_n$. It is shown that cyclic averages of equal powers
are the same for various $P_n(T_n)$, but their number is characteristic of
$P_n(T_n)$. Given the definition of a circle (sphere) by the vertices of
$P_n(T_n)$ and on the base of the cyclic averages are established the common
metrical relations of $P_n(T_n)$.
