Three major techniques are employed to calculate [pi]. Namely, (i) the perimeter of polygons inscribed or circumscribed in a circle, (ii) calculus based methods using integral representations of inverse trigonometric functions, and (iii) modular identities derived from the transformation theory of elliptic integrals. This note presents a combination of the first two procedures, which allows the derivation of a family of series that may exhibit very fast convergence rates. The geometrical interpretation gives good insight into the acceleration method that is being implemented.