The coboundary expansion generalizes the classical graph expansion to the case of the general simplicial complexes, and allows the definition of the higher-dimensional Cheeger constants $$h_k(X)$$ for an arbitrary simplicial complex X, and any $$k\ge 0$$ . In this paper we investigate the value of $$h_1(\Delta ^{[n]})$$ —the first Cheeger constant of a simplex with n vertices. It is known, due to the pioneering work of Meshulam and Wallach [12], that $$\begin{aligned} \lceil n/3\rceil \ge h_1(\Delta ^{[n]})\ge n/3, \text { for all } n, \end{aligned}$$ and that the equality $$h_1(\Delta ^{[n]})=n/3$$ is achieved when n is divisible by 3. Here we expand on these results. First, we show that $$\begin{aligned} h_1(\Delta ^{[n]})=n/3, \text { whenever }n\text { is not a power of }2. \end{aligned}$$ So the sharp equality holds on a set whose density goes to 1. Second, we show that $$\begin{aligned} h_1(\Delta ^{[n]})=n/3+O(1/n),\text { when }n\text { is a power of }2. \end{aligned}$$ In other words, as n goes to infinity, the value $$h_1(\Delta ^{[n]})-n/3$$ is either 0 or goes to 0 very rapidly. Our methods include recasting the original question in purely graph-theoretic language, followed by a detailed investigation of a specific graph family, the so-called staircase graphs. These are defined by associating a graph to every partition, and appear to be especially suited to gain information about the first Cheeger constant of a simplex.