We study the Reidemeister torsion and the analytic torsion of the m-dimensional disc in the Euclidean m-dimensional space, using the base for the homology defined by Ray and Singer in [10]. We prove that the Reidemeister torsion coincides with the square root of the volume of the disc. We study the additional terms arising in the analytic torsion due to the boundary, using generalizations of the Cheeger-Müller theorem. We use a formula proved by Brüning and Ma [1], that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary [6]. Some of our results extend to the case of the cone over a sphere, in particular we evaluate directly the analytic torsion for a cone over the circle and over the 2-sphere. We compare the results obtained in the low dimensional cases. We also consider a different formula for the boundary term given by Dai and Fang [4], and we show that the result obtained using this formula is inconsistent with the direct calculation of the analytic torsion.