We characterize projections among positive norm-one elements in unital \(\hbox {C}^*\) -algebras in purely geometric terms determined by the norm of the underlying Banach space. Concretely, let A be a \(\hbox {C}^*\) -algebra (or a \(\hbox {JB}^*\) -algebra) whose positive cone and unit sphere are denoted by \({A}^+\) and \(\textrm{S}_{A}\) , respectively. The positive portion of the unit sphere in A, denoted by \(\textrm{S}_{{A}^+}\) , is the set \({A}^+ \cap \textrm{S}_{A}\) , while the unit sphere of positive norm-one elements around a subset \(\mathscr {S}\) in \(\textrm{S}_{A^+}\) is the set \(\begin{aligned} \hbox {Sph}_{_{\textrm{S}_{{A}^+}}} (\mathscr {S}) :=\Big \{ x\in \textrm{S}_{{A}^+} : \Vert x-s\Vert =1 \hbox { for all } s\in \mathscr {S} \Big \}. \end{aligned}\) Assuming that A is unital, we establish that an element \(a\in \textrm{S}_{{A}^+}\) is a projection if, and only if, it satisfies the double sphere property, that is, \(\hbox {Sph}_{_{\textrm{S}_{{A}^+}}} \left( \hbox {Sph}_{_{\textrm{S}_{{A}^+}}} \left( \{a\}\right) \right) = \{a\}.\)
