We enumerate rooted 3-connected (2-connected) planar triangulations with respect to the vertices and 3-cuts (2-cuts). Consequently, we show that the distribution of the number of 3-cuts in a random rooted 3-connected planar triangulation with $n+3$ vertices is asymptotically normal with mean $(10/27)n$ and variance $(320/729)n$, and the distribution of the number of 2-cuts in a random 2-connected planar triangulation with $n+2$ vertices is asymptotically normal with mean $(8/27)n$ and variance $(152/729)n$. We also show that the distribution of the number of 3-connected components in a random 2-connected triangulation with $n+2$ vertices is asymptotically normal with mean $n/3$ and variance $\frac{8}{ 27}n$ .