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The workspace of a robot is an important criterion in evaluating manipulator geometry and kinematic performance. A systematical approach was presented in the paper to computing the shape and size of the workspace of planar robot manipulator. Firstly, a novel algorithm based on the Monte Carlo method was proposed to depict the boundary curves of robot workspace. Then, two methods for calculating area of the workspace were presented. One method based on numerical integration algorithm on closed curves. And the second method, the area can be estimated by probability and statistics. A planar robot manipulator consisting of 3 revolute joints was used as an example to illustrate the systematical approach.