Shape attribute acquisition is necessary in 3D point cloud registration, segmentation, recognition and other tasks. Traditional shape attributes are sensitive to scale changes, complicated to calculate，and simple to express the geometric meaning. Thus, combined with the concept of similarity dimension in fractal geometry, we propose a dimension definition that can be used as the shape attribute of the point cloud model. Firstly, we obtain a set of points that composed of the k neighborhood of each point in the model, and calculate the radius of the circumscribed sphere of the set of points. Secondly, we calculate the volume and area of the set of points, and solve the scale-sensitive problem by scaling. Finally, the dimension value of each point in the point cloud model is calculated using the similarity dimension expression, and the value is used to represent the shape attribute of the point cloud model. Experimental results show that the similarity dimension has the ability to express the shape and can clearly express the global characteristics of the model.