We use kinematic simulation (KS) to study the development of a material line immersed in a three-dimensional turbulent flow. We generalize this study to a material surface, cube and sphere. We find that the fractal dimension of the surface can be explained by the same mechanism as that proposed by Villermaux & Gagne (Phys. Rev. Lett. vol. 73, 1994, p. 252) for the line. The fractal dimension of the line or the surface is a linear function of time up to times of the order of the smallest characteristic time of turbulence (or Kolmogorov timescale). For volume objects we describe the respective role of the Reynolds number and of the object's characteristic size. Using the method of characteristics with KS we compute the evolution with time of a concentration field $C({\bm x},t)$ and measure the fractal dimension of the intersection of this scalar field with a given plane. For these objects, we retrieve the result of Villermaux & Innocenti (J. Fluid Mech. vol. 393, 1999, p. 123) that the Reynolds number does not affect the development of the fractal dimension of the iso-scalar surface and extend this result to volume geometries. We also find that for volume objects the characteristic time of development of the fractal dimension is the large scales' characteristic time and not the Kolmogorov timescale.