Not spread too thin

The trouble with being metaphorically spread too thin is that I end up too tired to cook properly, too tired to stumble through the day without calorific support, too tired to have the willpower to not eat the cheese, or the chocolate, or the peanuts, or the ice-cream. And the result of that is that there's rather more of me than I'd like. And the distinctly unfair thing is that, unlike my strength and psyche, it's not spread nice and thin, it's all determinedly lurking round my midriff. Not just piling on in a lardy tummy, but encircling me like a belt of purest butter. My face has gained no weight*, my arms are still reassuringly slender*, my legs are the same pins they ever were*. Even my bottom is remaining cheerfully the same size*. But my trousers are telling a nasty story about my waistline.

And because I am who I am, I thought I'd try and work out roughly what effect the extra weight would have it was spread thinly, compared to the way it's actually spread. And what's more, I thought it would be fun to do it without looking any data up, just to see if I could still do some good old fashioned "work it out from first principles" maths/science.

So first of all, I need to start with some approximations...

If I want to know how thin a layer the extra weight I'm carrying would make, I need to know my surface area. Hmmm, tricky... I'll assume I'm roughly cylindrical, with a height of 173cm and an average radius of 15cm. Some of me is definitely wider than that, and some definitely narrower, but then there's fiddly bits like arms and legs to consider, so I'll go with a generous average radius.

PhysicsBear approximates to a cylinder

A little bit of basic geometry and I know that my surface area is therefore, very roughly,

Surface Area	= (height × circumference) + (2 × area of end circle)
	= [h × (2 × π × r)] + 2 × π × r²
	= 17500 cm2

I'm also going to assume that any fat I'm carrying has approximately the same density as butter. And the standard block of butter I buy, I know weighs 250g, with dimensions of about 15cm x 3cm x 6cm.

Density of butter	= weight/volume
	= 250g / (15 × 3 × 6) cubic centimetres
	= 0.9 g/cc

At present, I'm about 3600g heavier than I want to be, or rather, 3600g heavier than I was when my trousers were comfortable.

Volume	= weight/density
	= 3600g / 0.9 g/cc
	= 4000 cc

In other words I'm carrying about 4000 cc of extra butter. If I were to spread that 4000 cubic centimetres evenly over my 17500 cm² surface, I'd add 4000/17500cm across my whole body. Or just over 2mm all over. And I'm sure my trousers would barely notice that. Much.

Instead however, I have what can only be described as a butter cummerbund. If all those 4000cc of "butter" really are spread in a doom-laden belt of lard around my middle, I wonder how big a belt it would be?

The calculation is a little more complicated this time. To work out the dimensions of the girdle of shame, I need to work out the dimensions of thin-me's midriff and then smear 4000cc of fat around it. Here you go, a diagram will help:

Finding the butter cummerbund

So thin-me has radius "b" and fat-me has radius "a". "c" is the approximate extent of the butter cummerbund. I'd guess "c" to be about 15cm. And we'll say that "b" is also about 15cm, as before. At which point we realise that diagram is not to scale, but carry on regardless.

The volume of a cylinder is the area of the circular end multiplied by the length of the cylinder, so:

Volume of thin-me midriff = π × b² × c = 10,600 cc

But we know that I've added 4,000 cc of extra fat, and just for now we're pretending I've piled it all on top of that nice svelte thin-me midriff, so we need to add:

Additional volume of fat = 4,000 cc

And by the simple expedient of adding these two numbers together, we can discover the volume of not-at-all-svelte-me around the middle:

Volume of fat-me midriff = 10,600 + 4,000 cc = 14,600 cc

but

Volume of fat-me midriff = π × a² × c

Therefore

π × a² × c = 14,600 cc

And thus we can calculate what we expect "a" to be if I have deposited all my extra weight in my hypothetical butter-band.

a	= √[14,600 / (π × c)]
	= 17.6 cm

A radius of 17.6cm is 2.6cm more than my estimate of thin-me. Which equates to approximately an inch in every direction around my waist. That's certainly what it feels like, though since that is actually the same as adding 16cm (>6") in circumference, I think it might not exactly be true. Or maybe some of my approximations are just a teensy bit off...**

I suspect I may have to admit that I do indeed have a gentle sprinkling of extra weight all over, with an unfortunate concentration about the waist. On the plus side, having flu has really kick-started the weight-loss.

* Each of these facts is indubitably untrue, but I like to deceive myself.

** Apparently there's a thing called the du Bois formula for calculating Body Surface Area:

BSA = 0.007184 × W^0.425 × H^0.725

And according to my own values of W (weight) and H (height), I come out as having a Body Surface Area of 17,594cm² which is disturbingly close to my hand-waving estimate of 17,500cm². If I didn't know better, I'd suspect myself of cheating.

Meanwhile, the density of butter is actually 0.911g/cc and the density of human fat is 0.9g/cc. More disturbing accuracy.