  In this paper, we analytically study the critical exponents and universal
amplitudes of the thermodynamic curvatures such as the intrinsic and extrinsic
curvature at the critical point of the small-large black hole phase transition
for the charged AdS black holes. At the critical point, it is found that the
normalized intrinsic curvature $R_N$ and extrinsic curvature $K_N$ has critical
exponents 2 and 1, respectively. Based on them, the universal amplitudes
$R_Nt^2$ and $K_Nt$ are calculated with the temperature parameter $t=T/T_c-1$
where $T_c$ the critical value of the temperature. Near the critical point, we
find that the critical amplitude of $R_Nt^2$ and $K_Nt$ is $-\frac{1}{2}$ when
$t\rightarrow0^+$, whereas $R_Nt^2\approx -\frac{1}{8}$ and
$K_Nt\approx-\frac{1}{4}$ in the limit $t\rightarrow0^-$. These results not
only hold for the four dimensional charged AdS black hole, but also for the
higher dimensional cases. Therefore, such universal properties will cast new
insight into the thermodynamic geometries and black hole phase transitions.
