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Body Mass Index
All things being equal (which they never are), the volume of a body goes up in proportion to the cube of a linear dimension. The volume of a sphere of radius 2r will be 8 times the volume of a sphere radius r. If the body has uniform density then the weight of the body will also increase as the cube of a linear dimension. So, in theory, a person who is 6ft tall and of a healthy weight should weigh 8 times as much as a person 3ft tall of a healthy weight. So why is the BMI worked out by dividing the weight of a person by the square of the height? I know that people do not have a uniform density, but I don't see why that should make much of a difference.
Answers
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To put it simply; a human body is more like a cylinder than a sphere so the mass varies as the square of the radius.
06:19 Mon 15th Feb 2016
when you got to the bit that a 3' man should weigh an eighth of a 6' man
why didnt you say to yourself
oops observation has refuted theory and so I will have to change my theory ?
in which case one way to go is to say to yourself ( again ) ah I wonder if the formula for BMI in fact the one I am criticising, is better than my own ?
anyway as the posters have noted it is all about SCALING
and the associated idea that you reduce the effect of some variable and can then see the effect of another lot ( of variables )
try goggling
scaling mathematical modelling
abd you will see that there is an awful lot on the internet about it
good books to read are
Growth and Form - d'arcy thompson
first edition 1914 and one of the few publication where people say well actually the first edition was better than the revised 3 rd edition - they said he hadnt scaled down his ideas - pun intended - and by the third edition the text had become sort of flabby
years and years ago I bought
scaling why is animal size so important ? scnmidt nielsen
as I was working in the childrens sector and intake and medication is done on a m2 basis of per kg and I wondered why - why the two and was there a difference.
which is almost what you are asking in your question
why didnt you say to yourself
oops observation has refuted theory and so I will have to change my theory ?
in which case one way to go is to say to yourself ( again ) ah I wonder if the formula for BMI in fact the one I am criticising, is better than my own ?
anyway as the posters have noted it is all about SCALING
and the associated idea that you reduce the effect of some variable and can then see the effect of another lot ( of variables )
try goggling
scaling mathematical modelling
abd you will see that there is an awful lot on the internet about it
good books to read are
Growth and Form - d'arcy thompson
first edition 1914 and one of the few publication where people say well actually the first edition was better than the revised 3 rd edition - they said he hadnt scaled down his ideas - pun intended - and by the third edition the text had become sort of flabby
years and years ago I bought
scaling why is animal size so important ? scnmidt nielsen
as I was working in the childrens sector and intake and medication is done on a m2 basis of per kg and I wondered why - why the two and was there a difference.
which is almost what you are asking in your question
An average man 2 metres tall ideal weight is between 74 and 100.4 kg [http://www.nhs.uk/tools/pages/healthyweightcalculator.aspx?tag=], the 'perfect' weight would be 87.2 kg. The average (or ideal) weight for a boy of 4 years is 16.3 kg with a height of 1.023 m.[http://www.disabled-world.com/artman/publish/height-weight-teens.shtml] Let's say 16 kg for 1 m. 16x8=128kg (Theoretical 'ideal' weight of man twice the height of a 1m boy.)
I would say that's not too far out (especially from the upper limit of the 'ideal' weight) from the cubic relationship. I didn't expect it to be a perfect match. It's almost as good a fit as a square relationship, which would give a weight or 64kg for a 2m-tall man.
In response to bhg481 (which I had put as best answer before I thought about it some more), OK the human body is more like a cylinder than a sphere, so why not divide the weight by the square of the waist measurement, rather than the square of the height? The mass of a cylinder would be proportional to the height (for a constant radius), not the square of the height.
I would say that's not too far out (especially from the upper limit of the 'ideal' weight) from the cubic relationship. I didn't expect it to be a perfect match. It's almost as good a fit as a square relationship, which would give a weight or 64kg for a 2m-tall man.
In response to bhg481 (which I had put as best answer before I thought about it some more), OK the human body is more like a cylinder than a sphere, so why not divide the weight by the square of the waist measurement, rather than the square of the height? The mass of a cylinder would be proportional to the height (for a constant radius), not the square of the height.
You're absolutely right Bert45, the radius of the person is more significant than the height. Bearing in mind that BMI is an approximate guide, perhaps the formula has been derived experimentally by comparing the height of "healthy" people with their weight. Taking your 3ft/6ft example, if a 3ft tall child weighed 5 stone I would expect a healthy 6ft adult to weigh 4 x 5 = 20 stones rather than 8 x 5 = 40 stones.
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