The Roman amphitheatre in Tarragona was built in the first half of the second century AD. We present a study of its formal layout based upon classical discussions of the construction of Roman amphitheatres through determination of the geometrical forms involved in their layout, ellipsis and ovum. The discussion considers the elevation of conical sections with the ellipse compared to the lowered forms derived from the circle with the oval - figures which are formally often confused. The question considered in this study - a determination of the elevation of the amphitheatre in Tarragona - is based upon a purely geometric analysis; we explicitly avoid considering the instruments required to construct the curve that creates the shape of the amphitheatre. A land survey of the area around the arena enabled us to establish the dimensions of its main axes, the curve described by the original remains of the podium, and the start of the seating tiers. Because the arena apparently has an elliptical layout, our approach to the problem involved determining the four- and eight-centred ovals that provided the best approximation for the only ellipse that fits into the axes that we determined, and we derive statistical confirmation from the deviations of the various figures from the physical reality of the building.