All the fish in a school occupy a volume estimated as N·BL3, where N is the number of fish and BL is their mean body length. We present extensive data from our experiments on cruising schools of saithe (Pollachius virens), herring (Clupea harengus) and cod (Gadus morhua) to validate this formula. Two methods of calculating the volumes of schools are described. One method is aggregative and depends on measuring the envelope of free space around a schooling fish, whereas the other is based on the dimensions of the school as a whole. The whole-school method is more reliable since it includes lacunae between the sub-units which exist in schools. For this method, we derive a computation which eliminates bias from outliers. The most realistic theoretical aggregative packing model predicts a volume per fish of 0.6 BL3. In saithe, the envelope of free space is approximately an ellipsoid, which, although it becomes more compressed at higher swimming speeds, yields a volume close to 0.7 BL3. From the whole-school method we calculate average volumes of 1.4 BL3 for saithe and 0.7 BL3 for herring. Increase in swimming speed produces more compact schools in saithe, but changes in arousal level can generate equally large differences. Changes in volume were not adequately explained by changes in nearest neighbour distance, giving support to the whole-school method.
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