In this mostly expository note, I give a quick proof of the definable Chow theorem of Peterzil and Starchenko using the Bishop-Stoll theorem and a volume estimate for definable sets due to Nguyen and Valette. The volume estimate says that any
d
d
-dimensional definable subset of
S
⊆
R
n
S\subseteq \mathbb {R}^n
in an o-minimal expansion of the ordered field of real numbers satisfies the inequality
H
d
(
{
x
∈
S
:
‖
x
‖
>
r
}
)
≤
C
r
d
\mathcal {H}^d(\{x\in S:\lVert x\rVert >r\})\leq Cr^d
, where
H
d
\mathcal {H}^d
denotes the
d
d
-dimensional Hausdorff measure on
R
n
\mathbb {R}^n
and
C
C
is a constant depending on
S
S
. Since this note is intended to be helpful to algebraic geometers not versed in o-minimal structures and definable sets, I review these notions and also prove the main volume estimate from scratch.