Articulatory speech synthesis based on aero-acoustic simulations of the vocal tract is computationally expensive and, therefore, requires simple yet precise models. Modeling the one-dimensional vocal tract area function directly instead of a higher dimensional vocal tract model is an efficient way to minimize the computational overhead of the simulations. In this paper, we propose a new parametric vocal tract model that is controlled by six points and capable of modeling a large variety of vocal tract shapes. We geometrically and perceptually evaluated the model on a set of 22 reference area functions corresponding to German vowels and consonants. The model was able to geometrically approximate the reference area functions with a minimum root-mean-square error of 0.302 cm $2$, a maximum error of 1.142 cm $2$, and a median error of 0.891 cm $2$. After optimizations, a perceptual evaluation of the synthesis using our model in combination with a state-of-the-art aero-acoustic simulation achieved a vowel recognition rate of 90.7% and a consonant recognition rate of 73.2%.