The measurement of vertebral heights is necessary for the evaluation of many disorders affecting the spine. High precision is particularly important for longitudinal studies where subtle changes are to be detected. Computed tomography (CT) is the modality of choice for high precision studies. Radiography and dual emission X-ray absorptiometry (DXA) use 2D images to assess 3D structures, which can result in poor visualization due to the superimposition of extraneous anatomical objects on the same 2D space. We present a semi-automated computer algorithm to measure vertebral heights in the 3D space of a CT scan. The algorithm segments the vertebral bodies, extracts their end plates and computes vertebral heights as the mean distance between end plates. We evaluated the precision of our algorithm using repeat scans of an anthropomorphic vertebral phantom. Our method has high precision, with a coefficient of variation of only 0.197% and Bland-Altmann 95% limits of agreement of [-0.11, 0.13] mm. For local heights (anterior, middle, posterior) the algorithm was up to 4.2 times more precise than a manual mid-sagittal plane method.