If the human body is stretched in three directions of width, length and height, the weight of a human body is proportional to the cube of its height. However, this is not valid in reality, and the body-mass index (BMI) is often calculated from the weight divided by the square of the height.



From such definition of the BMI, Prof. Beom Jun KIM has shown that the waist circumference of a human is not proportional to the height, but proportional to the square root of the height. In other words, if the body-mass index is the same, the taller person is thinner. “This is the reason why many fashion models are tall,” Prof. KIM says.



By analyzing the data of the length and weight of a variety of fishes, whales, and quadrupedal land mammals, Prof. KIM has shown that weight is proportional to the square of the height only for humans. In the case of other animals, the weight is proportional to the cube of the height. He presumed the reason why the calculation of the body-mass index of a human differs from that of other animals is that humans are bipedal, standing and walking upright.



From this, he predicted that in the case of infants who cannot walk yet, their weight should be proportional to the cube of their height, but children older than about one year old should have a weight proportional to the square of their height. This prediction was tested for data from Sweden, Korea, and the World Health Organization, and was confirmed to be correct. If we consider the human body as the form of a simple cylinder and apply the condition that the torque by gravity and the torque by muscle must be in balance, we can show that the weight of a human should be proportional to the square of the height using Newtonian mechanics of physics.



Prof. KIM also said that if we measure the size of the pelvis and the height from human fossil records, we will be able to deduce the time when mankind started to walk upright.

Figure 1. Graphs of the relationship between the height (H) and the weight (M) of pale chub fish, whales and land mammals. All three graphs show that the animals’ weight is proportional to the cube (p = 3) of the height. For land animals, the height (H) can be measured in two ways: shoulder height and head-to-tail length. Both ways to measure the height meet the relationship (M is proportional to the cube of H).

Figure 2. Graphs of the relationship between the height (H) and the weight (M) of children from Sweden and Korea. In this figure, the purple dots represent infants younger than one year old, and the orange dots represent children older than one year old. At one year of age, the time when a child begins to walk, the relationship between the height and weight is changed. Before one year, the weight is close to the cube of the height as with the other animals in Figure 1 above, but after one year, weight is close to the square of the height.

Figure 3. A simple model of a human body. By applying the equilibrium condition of the torque by gravity (Fg) and the torque by muscle (Fm), we can show that the weight of a human being should be proportional to the square of the height.

Figure 4. According to the body-mass index calculation method, the waist circumference of a human being is proportional to the square root of the height. The figure shows how body somatotype is changed according to the height if the body-mass index is the same. The taller the person is, the thinner it becomes. In the figure, when the height doubles, the waist circumference is about 1.4 times (or the square root of 2) smaller.

Figure 5. Unlike a human body, when the length of a fish doubles, the height of the torso also doubles. In other words, it is difficult to know the actual fish size only from fish pictures without comparisons because both big fish and small fish look the same. This is because the weight (or volume) of the fish is proportional to the cube of its length.