Certain relationships between the Newtonian gravitational constant, the Planck constant, and the square of the fine structure constant, established by dimensional analysis, are presented. Here we show that, based on these relationships, a more exact value for the Newtonian gravitational constant G equal to 6.67409076 × 10−11 m3 kg−1 s−2 can be calculated. In this way, these relationships could be used as a nonconventional tool for establishing a G gravitational constant value very close to the real one. It is considered that the difference between this calculated value and the values provided by the most accurate measurements of this constant is very important, whereas such difference could reflect certain, subtle and unknown “links” existing between the natural phenomena. This article... also highlights a very interesting relationship between the Newtonian gravitational constant G, the square of the fine structure constant (α−1)2, and the Planck constant h, having the following form: 2XG = π (10Xα/2Xh)2, where XG, 10Xα, and Xh are the normalized values (dimensionless) of these constants.