The shapes of excluded volume, two-dimensional star polymers are explored with the renormalization group and Monte Carlo computer simulation of a tangent hard disk model. The mean-square radius of gyration, the g-ratio, and the asphericity of linear and star polymers are examined. The standard c expansion for the g-ratio is modified to use the known value of the radius of gyration expansion exponent rather than only the first-order expansion value. Good agreement is obtained with the current Monte Carlo Pivot algorithm results and previous simulations of other polymer models. Our approach also improves the predictions of the g-ratio in three dimensions, significantly enhancing the accuracy of the renormalization group for calculating shape properties.