  We prove that there is an absolute constant $ C$ such that for every $ n \geq
2 $ and $ N\geq 10^n, $ there exists a polytope $ P_{n,N} \subset \mathbb{R}^n
$ with at most $ N $ facets that satisfies
$$\Delta_{v}(D_n,P_{n,N}):=\text{vol}_n\left(D_n \Delta P_{n,N}\right)\leq
Cn^{-2/(n-1}\text{vol}_n\left(D_n\right)$$ and
  $$ \Delta_{s}(D_n,P_{n,N}):=\text{vol}_{n-1}\left(\partial\left(D_n\cup
P_{n,N}\right)\right) - \text{vol}_{n-1}\left(\partial\left(D_n\cap
P_{n,N}\right)\right) \leq 4CN^{-\frac{2}{n-1}} \text{vol}_{n-1}\left(\partial
D_n\right),
  $$ where $ D_n $ is the $ n$-dimensional Euclidean unit ball.
  This result closes gaps from several papers of Hoehner, Ludwig, Sch\"utt and
Werner. The upper bounds are optimal up to absolute constants. This result
shows that a polytope with an exponential number of facets (in the dimension)
can approximate the $ n$-dimensional Euclidean ball with respect to the
aforementioned distances.
