We give a new and clever proof of the Russo-Dye theorem for JB*-algebras, which depends on certain recent tools due to the present author. The proof given here is quite different from the known proof by J. D. M. Wright and M. A. Youngson. The approach adapted here is motivated by the corresponding C*-algebra results due to L. T. Gardner, R. V. Kadison and G. K. Pedersen. Accordingly, it yields more precise information. Incidentally, we obtain an alternate proof of Russo-Dye Theorem for C*-algebras. A couple of further results due to Kadison and Pedersen have been extended to JB*-algebras as corollaries to the main results.