# The FugueCounterpoint by Hans Fugal

20Aug/077

When you try to lose weight, what you are really trying to do is lose fat.
Weighing yourself is a first approximation of your progress, but a better
indicator is your body fat percentage (%BF). Unfortunately, measuring %BF can
be expensive and/or difficult. It doesn't need to be so. All you need is a body
of water (e.g. a swimming pool), a gallon jug, and your bathroom scale.

One of the most accurate ways to measure body fat is hydrostatic weighing. You are weighed underwater and on land, and your body's density is determined. Then body fat is estimated from the measured density. This is the same basic technique that we will use, but we don't require an underwater scale or special tank.

First the how, then I'll give you the physics. Get in the pool and exhale all
the air you can, and allow yourself to sink. You will sink unless you're
particularly obese. Take note of the sensation of sinking. Then do the same
thing but with your lungs full. Take note of the sensation of floating. Now, we
want to reach the point of neutral buoyancy when your lungs are empty, where
you are neither sinking nor floating. You will be weightless under the water.
Take the gallon jug and hold it under the water, then exhale completely. If the
jug is full of air, you will probably float (unless you are quite lean, in
which case you'll need two jugs). Keep adding water and repeating until you
reach neutral buoyancy. If you sink, add air (pour out some water). If you
float, add water. Once you've found the magic amount of water, use this
equation to calculate your body density (ρ):

$\rho=\frac{m}{\frac{m}{\rho_w}-v_a}$

where m is your mass (what the scale tells
you), v is the volume of you and your buoy combined, and vair is the
volume of air in your buoy. If you have ¾ gallon of water in your gallon jug,
then vair is ¼ gallon. ρw=1 kg/liter for all the
precision we need.

Once you have density, you may like to estimate your body fat. The equation for that is Siri's equation, which says

$\text{BF}=\left(\frac{4.95}{\rho}-4.50\right)100$

This equation assumes your lungs are completely empty, which they can't be, so we need to introduce a term for the residual volume of your lungs. This is about ¼ of your total lung capacity, or ⅓ of your vital lung capacity. You can measure your vital lung capacity with a balloon or a by blowing air through a straw into an inverted container filled with water. The average residual lung capacity for an american male is 1.2 liters; mine is about 1.9 liters. So we can adjust the formula for density as follows:

$\rho=\frac{m}{\frac{m}{\rho_w}-v_a-v_r}$

If you do this experiment you will probably find that your estimated %BF is not too far from your BMI, which is a statistical tool for estimating %BF. It can be wildly inaccurate for statistical outliers (e.g. people who are actually in shape), but it's easy to calculate and a decent sanity check in this case.

Here's what's going on. We're using Archimedes' principle: the buoyant force on a submerged object is equal to the weight of the fluid displaced. When the buoyant force balances the force of gravity, we have neutral buoyancy. The buoyant force is expressed as $F_b=-\rho_w v g$. Substitute weight for the buoyant force and solve for the volume of the body (v = vbody + vair), then substitute that into the definition of density (m/v), and you get the formula I gave you above (if you consider the mass of air and the gallon jug as negligible). I glossed over that—if you'd like me to go into more detail say so in the comments.

If you're particularly obese and don't sink when you exhale completely, then all is not lost. You just need some counterbalance. The modified equation is:

$\rho=\frac{m_b}{\frac{m_b+m_c}{\rho_w}-v_a-v_r-v_c}$.

You can find the volume of your counterbalance by taking a cue from Archimedes and measuring displacement.

About accuracy: the biggest variable in this process is how much air is left in your lungs. You will find with practice that you are able to exhale more air, which will lower your %BF estimation, as if by magic. However it always overestimates and once you figure out how to completely exhale will be very consistent. Siri's equation is the next place to look—it basically takes the density of fat and the density of muscle and ignores bone mass and density, what you ate for lunch, etc. It will also almost certainly overestimate %BF. The astute reader will wonder about air compression in the milk jug. I measured this and found that when the jug is held within a foot or so from the surface, it does compress. However, the amount it compresses conveniently offsets the extra capacity of the jug (they don't pack milk spilling over the brim of the jug, after all). All in all I think it's accurate within a few percentage points for %BF, gives you an upper bound (i.e. you are free to brag about the number you get, even if it may be slightly high), and is more accurate than BMI.

1. Very interesting. This morning I went to the pool with the kids and tried your method while they were swimming their workout laps.

I calculated my body fat percentages using this method of hydrostatic weighing and compared the results to the skinfold measurement method and the BMI calculation. Here are the results:

Weight – 111.7 Kg
Height – 1.93 m
BMI – 30.0%
Skinfold – 29.0%
Hydrostatic – 25.3%

I had previously noted that skinfold measurements gave a lower bodyfat reading than the BMI. I expected this since BMI does not take into account higher than average muscle mass. I expected to see a closer correlation between the skinfold method and hydrostatic weighing. The skinfold method is an empirical relationship derived by comparing the total readings (in mm) of three body sites with hydrostatic weighing results. A curve is then fit to the data points and an empirical equation derived from the curve. This process is repeated for different ages since the relative percentage of subcutaneous fat changes with age. I don’t have the equation; the book I have simply has nomographs to solve the unstated equations.

So, for the \$64 question, “Which of my two measurements was more accurate?” I don’t know. The vague estimate of residual lung capacity bothers me a little. How accurate is this? and How do the “profesionals” measure this variable in clinical hydostatic weighing?

On the other hand, the skinfold caliper measurements are highly variable depending on technique. And even with consistent technique by the same person the readings still vary by a few millimeters on each measurement which theoretically could compound to an varience of up to 2+ percentage points in the result. Errors in exactly how and where to measure could give errors in the result that are even more dramatic. It is my observation that the thicker the subcutaneous fat layer, the more difficult it is to get consistent measurements.

Given the above observations, I tend to believe that my hydrostatic calculated result is the more accurate. However, given the significantly lower result, I question whether it really will “almost certainly overestimate %BF”.

Thanks for the post. It was interesting to make the comparisons.

2. Hans,

I was directed to this blog from searching the body mass topics, but have since read quite a few of your posts. All very interesting stuff. I think I share the same completely random interests as you. In this blog I cannot pull up the all important body density calcuation in the first white block. Could you forward to me or post here.

Thanks,

Paul Stone

3. Sorry Paul, migrating blog software had messed up some things (and the formatting still leaves something to be desired). Hope it’s clearer now.

4. Hello,

This is a message for the webmaster/admin here at hans.fugal.net.

Can I use some of the information from your blog post above if I provide a link back to your website?

Thanks,
William

5. Sure, William

6. Hi Hans,

I found your blog because i’d like to settle an argument and learn something in the process. My partners weight is comparatve to my own. I say comparative beacause if i actually say she will kill me! I am learning to swim and she makes treading water look effortless and I seem to have more difficulty than her so I’m trying to find out why? I exercise in the gym using weights 2 – 4 days per week and running etc I think the density of my body may have something to do with it but I’d just like to know your thoughts before I go Swimming again in two weeks whether my argument will win.

Just started blogging myself and share more details at my own blog

7. Trevor, if your weight and fitness are comparable then he will have more body fat by virtue of being female. So her body has a bit lower average density and will be more buoyant. There’s also technique, specific strength, and even the shape of feet/legs that could make an even bigger difference in experiences treading water. As a young lifeguard I was skinny and not the most fit by far, but I have wide feet, I am tall, and have a pretty good breaststroke kick. I could blow everyone out of the water in the times tread with a 5lb brick.