Several radii problems are considered for functions $f(z)=z+a_2z^2+...$ with fixed second coeffcient $a_2$. For $0\leq \beta<1$, sharp radius of starlikeness of order $\beta$ for several subclasses of functions are obtained. These include the class of parabolic starlike functions, the class of Janowski starlike functions, and the class of strongly starlike functions. Sharp radius of convexity of order $\beta$ for uniformly convex functions, and sharp radius of strong-starlikeness of order $\gamma$ for starlike functions associated with the lemniscate of Bernoulli are also obtained as special cases.