In this paper, we obtain the
L
p
L^p
restriction estimates for the truncated conic surface
Σ
=
{
(
ξ
′
,
ξ
n
,
−
ξ
n
−
1
⟨
ξ
′
,
N
ξ
′
⟩
)
:
(
ξ
′
,
ξ
n
)
∈
B
n
−
1
(
0
,
1
)
×
[
1
,
2
]
}
\begin{equation*} \Sigma =\big \{(\xi ’,\xi _n,-\xi _n^{-1}\langle \xi ’,N\xi ’\rangle ): (\xi ’,\xi _n)\in B^{n-1}(0,1)\times [1,2]\big \} \end{equation*}
with
N
=
I
n
−
1
−
m
⊕
(
−
I
m
)
N=I_{n-1-m}\oplus (-I_m)
for
m
≤
⌊
n
−
3
2
⌋
m\leq \lfloor \tfrac {n-3}2\rfloor
provided
p
>
2
(
n
+
3
)
n
+
1
p>\tfrac {2(n+3)}{n+1}
. The main ingredients of the proof are the bilinear estimates of strongly separated property and a geometric distribution about caps.