Additional Math Pages & Resources

Wednesday, January 12, 2011

How Fat Am I? Part III

Welcome to the Excel Math blog where I talk about things grown-ups can do with math they learned in elementary school - the kind of math our curriculum helps present to K-6 kids.

My wife is a PE teacher. For 35 years I have watched her measure people and calculate their fitness and fatness. Why does she do this?

We want to know how much of the volume of our body is made up of fat, and how much is solid material (bone), muscle tissue, water and air. With this information, we have a good idea of how healthy, strong, long-lived, etc. we are likely to be. Fitness professionals can then provide you with suggestions on how to improve your health.

My wife knows how to use a skin-fold caliper and can poke and pinch you until the truth emerges about your sub-cutaneous (under the skin) body fat. For many years, this was the most cost-effective and accurate way to determine body composition. The problem is, it's hard for an amateur to do accurately, and almost impossible to measure yourself. And the results vary for children, adults, and various ethnic groups.

This morning I had my wife take some measurements on me using her simplest skin-fold caliper. Here she is pinching and measuring my waist while I take the photos.

Notice the caliper has two arrows - you are supposed to press with your thumb until they line up - this ensures you close the caliper adequately. A plastic arm deflects when the jaws are tight enough on your fat. The settings seem a bit tight to me - ouch!

In the second picture she is measuring at the back of my upper arm. The scale indicates mm of body fat. These are the only two places we measured, but there are many more that can be checked. Go here to see some examples.

If this was a real test, she would have made multiple measurements around my body. Then using software or a calculator, she would add all the measurements (in mm) and calculate as shown below. This is called the Jackson & Pollock equation:

MALE Body Density = 1.0990750 - 0.0008209 (X2) + 0.0000026 (X2)2 - 0.0002017 (X3) - 0.005675 (X4) + 0.018586 (X5)
X2 = sum of the chest, abdomen and thigh Skinfold in mm 

X3 = age in years 
X4 = waist circumference in cm 
X5 = forearm circumference in cm

FEMALE BD = 1. 1 470292 - 0.0009376 (X3) + 0.0000030 (X3)2 - 0.000 1 1 56 (X4) - 0.0005839 (X5)
Where X3    = sum of triceps, thigh and suprailiac Skinfold, in mm 

X4 = age in years
X5 = gluteal circumference, in cm

Then using the Siri equation, she would convert body density to % of body fat, like this

FAT% = ((4.95 ÷ Body Density) - 4.5) x 100

Does that seem like a lot of work? Yes, indeed it does. But it's elementary math - nothing that the average person couldn't handle with a little care and attention to the decimal places.

You might wonder how these obscure formulas were developed. People did thousands of these skin fold tests trying to find a correlation to the ultimate body composition method - hydrostatic weighing.

Before we talk about hydrostatic weighing on Friday, we'll look at another common method known as bioelectrical impedance analysis. Sounds impressive. It's a special scale that you step on, and it gives your weight and an estimate of your body fat.


  1. I used your formulate:
    Body Density = 1.0990750 - 0.0008209 (X2) + 0.0000026 (X2)2 - 0.0002017 (X3) - 0.005675 (X4) + 0.018586 (X5)
    I'm a male with X2 = 26 (mm from 6+10+10) & X2 = 676 (mm^2), x3=3 53 (y.o.), x4 = 78.1 (cm) and x5 = 27.3 (cm) so I get BD of 1.1330493
    This number appears to be too big and I get negative % Body Fat using Siri Formula (495/BD)-450. Can you comment on why I end up with a negative % Body Fat? Thank you!

    1. Hi Peter, Thanks for the comment. It appears your X3 should equal 53. Even so, I come up with the same numbers you do. This post was written over a year and a half ago, and the author has now retired and moved on to other endeavors (such as painting and collecting religious icons).

      Perhaps someone else can help us resolve this issue with these formulas. In the meantime, here's an online calculator you can use to plug in the numbers of your choosing. It doesn't use the Jackson & Pollock equation so it will give you a body density of 1.08 and a body fat percentage of 10.33%. The SIRI formula is the second body fat calculation shown:

      Wishing you all the best with your calculations!


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