Let $\alpha_m$ be the Lebesgue measure of the closed unt ball in $\mathbb{R}^m$. For $A \subset \mathbb{R}^n$, the $m$-dimesnsional Hausdorff measure of $A$ is defined as: $$ H^m(A) = \lim_{\delta \to 0 } \inf \bigg\{\sum_j \alpha_m \bigg(\frac{1}{2}\bigg)^m (\text{diam}(S_j))^m \bigg| A \subset \bigcup_j S_j, \ \text{diam}(S_j) < \delta \bigg\} \quad \quad (*) $$
I'm trying to understand this definition and how the various components of it are motivated.
Suppose $n=m=3$ and that $A$ is the the closed unit ball in $\mathbb{R}^3$. From what I've read if $m=n$ the Hausdorff measure agrees with the Lebesgue measure so $(*)$ should evaluate to simply $\alpha_m = \alpha_n = \alpha_3 = \frac{4}{3}\pi$. How can this evaluation be performed starting from the definition of the Hausdorff measure in $(*)$?