In this manuscript, a pentagon shaped plasmonic graphene nano patch antenna is designed and investigated on silicon dioxide, silicon nitride and zinc oxide substrates. The optical characteristics of the graphene such as conductivity and permittivity are analyzed at various values of chemical potential using Kubo conductivity model. The plasmonic resonance of the proposed nano-antenna is analyzed through the graphene chemical potential varied in the span from 0.0 eV to 2.0 eV. It is observed that the proposed pentagon shaped graphene nano patch antenna shows good plasmonic resonance characteristics at 29.8750 THz with the reflection coefficient of -27.4153 dB, 91.5650 THz with -27.5179 dB and 853.7350 THz with -40.3267 dB on silicon dioxide, zinc oxide and silicon nitride substrates respectively. These optimum characteristics are observed at the chemical potential values 2.00 eV and 0.50 eV for the proposed graphene nano patch antenna.