In this work we study the thermodynamics of a (2+1)-dimensional static black hole under a nonlinear electric field. In addition to standard approaches, we investigate black hole thermodynamic geometry. We compute Weinhold and Ruppeiner metrics and compare the thermodynamic geometries with the standard description for black hole thermodynamics. We further consider the cosmological constant as an additional extensive thermodynamic variable. For thermodynamic equilibrium in three-dimensional space, we compute heat engine efficiency and show that it may be constructed with this black hole.