Here we use in toto imaging together with computational segmentation and analysis methods to quantify the shape of every cell at multiple stages in the development of a simple organ: the notochord of the ascidian Ciona savignyi. We find that cell shape in the intercalated notochord depends strongly on anterior-posterior (AP) position, with cells in the middle of the notochord consistently wider than cells at the anterior or posterior. This morphological feature of having a tapered notochord is present in many chordates. We find that ascidian notochord taper involves three main mechanisms: Planar Cell Polarity (PCP) pathway-independent sibling cell volume asymmetries that precede notochord cell intercalation; the developmental timing of intercalation, which proceeds from the anterior and posterior towards the middle; and the differential rates of notochord cell narrowing after intercalation. A quantitative model shows how the morphology of an entire developing organ can be controlled by this small set of cellular mechanisms.