Let m_1, m_2, ..., m_n be distinct primitive elements of a two-dimensional lattice M such that det(m_i, m_(i + 1))>0 for i = 1, ..., n - 1. Each collection Γ = {m_1, m_2, ..., m_n} then forms a set of rays of a unique complete fan in M, and therefore determines a two-dimensional toric variety X_Γ.