Few theorems are known about planar graphs. For, example, Kuratowski proved that a graph is planar if and only if it has no subgraph homeomorphic to K₅ or K₃,₃. It has remained as a direct criterion for determining whether a graph is planar or not. Powerful as the theorem is, it is not always easy to apply. This leads us to try some practical methods to test a planar graph. In this thesis, we have an algorithm for finding an outer circuit for a simple connected planar graph. Then, we use this outer circuit to draw a straight line graph in the plane. The programme for this algorithm is written in FORTRAN for an IBM 360/67 Computer.