  The Tur\'an number of a graph $H$, denoted by $\text{ex}(n, H)$, is the
maximum number of edges in an $n$-vertex graph that does not have $H$ as a
subgraph. Let $TP_k$ be the triangular pyramid of $k$-layers. In this paper, we
determine that $\text{ex}(n,TP_3)= \frac{1}{4}n^2+n+o(n)$ and pose a conjecture
for $\text{ex}(n,TP_4)$.
