RRT* is a recent and improved variant of the RRT path finding algorithm. While RRT concentrates on simply finding an initial obstacle-free path, RRT* guarantees eventual convergence to an optimum, collision-free path for any given geometrical environment. On the other hand, the main limitations of RRT* include its slow processing rate and high memory utilization due to the large number of iterations required to achieve optimal path solution. This paper presents Triangular Geometerised-RRT* (TG-RRT*) which incorporates Triangular geometrical methods in the RRT* algorithm and improves its processing time by decreasing the number of iterations required for optimal solution. Simulation results under different environments demonstrate an improved convergence rate of TG-RRT*, in comparison with RRT*.