Surface area of cube with size $a$ is multiple of $a^2$. The same holds for balls and any three dimensional object.

Volume of cube is $a^3$. Volume of ball and other three dimensional object is proportional to $a^3$. The mass of any object is its average density multiplied by its volume, hence also multiple of $a^3$.

Let's imagine ideally shaped human figure. If you scale it to different sizes leaving its relative proportions and density unchanged then its volume will be its height to the power of tree multiplied by a constant of ideal figure, the same holds for mass it is only multiplied by the density. The fraction $\frac{mass}{{height}^{3}}$ will be constant.

I am very surprised that the BMI is computed as $\frac{mass}{{height}^{2}}$ instead of $\frac{mass}{{height}^{3}}$. Why is it so?

My only explanation is that the formula was invented by some non mathematician who tried to compute $\frac{mass}{{height}^{1}}$ and sow that it isn't of much use so he replaced it by $\frac{mass}{{height}^{2}}$ which works better since human heights do not vary too much. But wouldn't be $\frac{mass}{{height}^{3}}$ much better?

Edit: I Just found that the Corpulence Index (CI) or Ponderal Index does the think I suggest. But my question remains. Why is BMI much more popular than the more correct index?

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    $\begingroup$ "My only explanation is that the formula was invented by some non mathematician" - Do you really think everyone else is just an idiot who doesn't know math? Not the best way to get around in the world I think. $\endgroup$ – Bryan Krause Jan 30 at 1:41
  • $\begingroup$ The problem is that a =/= height. Volume is a^3, but not height^3. $\endgroup$ – Nicolai Jan 30 at 16:30
  • $\begingroup$ I don't have time to post a proper answer, but you can read the relevant paper (if you have access) here: sciencedirect.com/science/article/pii/0021968172900276 $\endgroup$ – Jack Aidley Jan 31 at 14:44

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