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Body Math and Fitness Formulas

Updated on November 07, 2016
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TR Smith is a product designer and former teacher who uses math in her work every day.

Joined: 5 years agoFollowers: 215Articles: 456

When starting a new weight loss plan or workout routine, it's helpful to track your progress with different measures of fitness. The fitness indicators below use height, weight, age, heart rate, skinfold thickness, and other body measurements to show you what you need to work on to get in shape. The only tools needed are a scale, tape measure, pair of calipers, and calculator.

You can also program all of the formulas in to Excel to create a fitness tracking spreadsheet. Or, if you don't like doing the math, you can just use various online calculators to compute all the values. To skip all of the explanations for the formulas, scroll down to the bottom of this article for a complete list of equations.


BMI -- Body Mass Index -- is one of the most commonly used and overused measures of general health. It's simple to calculate as it only requires your weight and height. Not only is it easy to compute, but also easy to interpret; very low BMIs correspond to being underweight, very high BMIs correspond to being overweight or obese, and healthy people are somewhere in the middle. BMI is ubiquitous; even the World Health Organization uses it in famine statistics.

But, like anything that is too easy, there are drawbacks. One major disadvantage of BMI is that it doesn’t take into account body composition and distribution of mass. Obviously, since muscle is heavier than fat, body builders have high BMIs, while a fatty person could actually have a surprisingly low BMI. Where body mass is distributed also affects health. The picture shows 8 women who all have BMI values of 30, but different body types.

Body Mass Index is calculated by this formula for height in inches (H) and weight in pounds (W):

BMI = (703)(W)/(H²)

Or use this metric formula for weight in kilograms and height in meters:

BMI = W/(H²)

And this is the scale used to interpret it:

  • < 18.5 underweight
  • 18.5-24.9 healthy
  • 25-30 overweight
  • > 30 obese

Even though BMI doesn’t take into account anything besides weight and height, in the absence of other measurements, it can be used to roughly approximate other indicators of health, such as lean body mass and basal metabolic rate.

blue = underweight, green = healthy weight, yellow = overweight, red = obese
blue = underweight, green = healthy weight, yellow = overweight, red = obese

%BF and LBM

Three interrelated health measurements are total body fat, percent body fat (%BF), and lean body mass (LBM). Total body fat is the total weight of fat in your body. Percent body fat is the percentage of your weight that is fat. And lean body mass is the total weight of non-fat tissue in your body. For example, if a woman weighs 140 lbs and has 30 lbs of fat, then all of the following are equivalent:

  • total body fat = 30 lbs
  • % body fat = 30/140 = .214, or 21.4%
  • lean body mass = 140 – 30 = 110 lbs

There are many ways to compute the numbers. The traditional technique of determining how much fat a person has is by hydrostatic testing. Your body is submerged in a tank of water and your buoyancy is used to determine percentage body fat. The principle behind this method is the fact that fat floats in water. So, the more buoyant you are, the higher your percent body fat. This also gives the truest estimate of % body fat.

In lieu of of hydrostatic measurement (which is costly) you can also compute body fat with these methods.

Skinfold Analysis: Thickness of subcutaneous fat (the fat under your skin) is another predictor of fat. The traditional way to do this is to use calipers to record the thickness of folds on seven areas of the body, add them all up, and then plug that sum into an equation.

Girth Methods for Men: For men, body fat and lean body mass can be estimate by the circumference of the waist and neck, relative to weight and height. The simplified formula for men is

LBM = 94.42 - 4.15X + 1.082Y</blockquote>

where X is the waist measurement in inches and Y is the weight in pounds. This gives your lean body mass in pounds.

Girth Methods for Women: You can also estimate female body and fat and LBM with waist, wrist, hip, and forearm circumferences. The approximation formula for women is

LBM = .732Y - .157X + .318C - .249D + .434E + 8.98

where Y is weight in pounds, X is waist measurement, C is wrist measurement, D is hip measurement, and E is forearm measurement. Again, this formula returns lean body mass in pounds.

All these formulas were developed and tested by physiologists and health scientists. When the equation results are compared to the hydrostatic testing results, they are often accurate within 3%.


BMR stands for Basal Metabloic Rate. Your metabolic rate, aka metabolism, is the rate at which you burn calories as you expend energy throughout the day. Having a high metabolism is a great from a weight loss standpoint, since it means your workouts burn a lot of calories. A lower metabolism is not such a good thing for losing weight, since it means more exercise is needed to burn off calories. The golden rule for weight loss is that you must burn more calories than you consume.

The calories we see on food labels are actually kilocalories, units of 1000 calories, but in common speech, everyone calls kilocalories “calories." For example, the package of cookies that says "200 calories" is actually 200 kilocalories equivalent to 200,000 real scientific calories.

So what is the basal metabolic rate? Your BMR is the number of calories you would burn in a day if you did absolutely nothing but sleep for 24 hours. No moving, no snoring, no dreaming. It’s the number of calories a comatose person would need to maintain his current weight. Of course, people who are awake for part of the day will need to eat more calories than their BMR value. How much more depends on their activity level.

Let’s say that your BMR is X calories. Then the number of calories you need to take in is X multiplied by a factor between 1.2 and 1.9. 1.2 represents the low end of the activity scale, and 1.9 the upper end. Here are two formulas for estimating BMR based on sex, age in years (A), weight in lbs (W), and height in inches (H).

male BMR = BMR = 6.24W + 12.71H - 6.76A + 66.47
female BMR = 4.34W + 4.7H - 4.68A + 655.1

An alternative formula for calculating BMR uses your lean body mass and works for either sex:

BMR = 370 + 9.795*LBM

For example, let's say your BMR is 1450 calories per day. If your activity level is 1.3 on a scale of 1.2 to 1.9, then the number of calories you need to eat each day is 1.3*1450 = 1885.

Target Heart Rate

In order to burn calories and lose weight more efficiently, you have to exercise at the right intensity level. Too low, and you won’t get an effective work out or burn fat reserves. Exercising at an extremely high intensity is great if you are already in shape, but it won’t effectively burn fat if you are just beginning a regimen and have a lot of pounds to shed. Heart Rate Training Zone calculations use your resting pulse to find the optimum intensity of exercise for you.

While you are working out you can check your pulse to see if you are exercising at the right level. When you are exercising at 60%-85% of the maximum intensity level, your heart rate is said to be in the “training zone.” The training zone is a range of heart rates; the upper and lower ends of the range depend on your maximum heart rate (MHR) and your resting heart rate (RHR) measured in beats per minute (bpm).

The most commonly used formula for computing training zone heart rate is

Upper End = 0.85*MHR + 0.15RHR
Lower End = 0.6*MHR + 0.4*RHR

where MHR is your maximum heart rate in beats per minute (bpm) and RHR is your resting heart rate in bpm. RHR is found by taking your pulse when at rest, and MHM is estimated from the Karvonen formula:

MHR = 220 - Age

For example, suppose you are 27 with a resting heart rate of 57 bpm. Your MHR is 193 bpm and the upper and lower ends of your training zone heart rate are 172.6 and 138.6 respectively.

List of Equations

Here is a summary of equations for computing BMI, LBM, BMR, and target heart rates. Several variants are also included so you can compute measures of fitness with different formulas developed by different researchers.

  • BMI = 703W/H² [ H=inches and W=pounds ]
  • BMI = W/H² [ H=m and W=kg ]
  • male body density = 1.112 - (0.000435)F + (0.00000055)F² - (0.000288)A
    [ F = skinfold thickness in mm, A = age in years ]
  • Navy method male body density = 1.01774 - .19077*Ln(W-N) + .15456*Ln(H)
    [ W, N, & H =waist girth, neck girth, & height in inches, Ln is the natural logarithm ]
  • female body density = 1.097 - (0.00047)F + (0.00000056)F² - (0.000128)A
    [ F = skinfold thickness in mm, A = age in years ]
  • male lean body mass in pounds, LBM = 94.42 - 4.15X + 1.082Y
    [ X=waist inches, Y=weight lbs ]
  • female lean body mass in pounds, LBM =
    0.732A - 0.157B + 0.318C - 0.249D + 0.434E + 8.987
    [ A=weight lbs. B, C, D, & E = waist, wrist, hip, & forearm inches ]
  • LBM estimated with BMI
    LBM men = (1.1)W[1 - (.011636)BMI]
    LBM women = (1.07)W[1 - (.013832)BMI]
    [ W=weight, any units ]
  • Total Body Fat + LMB = Total Weight
  • LBM = Weight x (1 - P), P = % Body Fat, as a decimal
  • % Body Fat = (Total Body Fat)/(Total Weight)
  • % Body Fat Men = 495/(Body Density) - 450
  • % Body Fat Women = 496/(Body Density) - 451
  • male BMR = BMR = 6.24W + 12.71H - 6.76A + 66.47
    [ W=weight lbs, H=height inches, A=age years ]
  • female BMR = 4.34W + 4.7H - 4.68A + 655.1
    [ W=weight lbs, H=height inches, A=age years ]
  • Alternative BMR formula
    BMR m= (4.5306)W + (15.875)H - (4.92)A + 5
    BMR f = (4.5306)W + (15.875)H - (4.92)A - 161
    [ W=weight lbs, H=height inches, A=age years ]
  • BMR male or female = 370 + (9.795)LBM
    [ LBM in lbs ]
  • 4 Maximum Heart Rate Formulas
    MHR = 220 - A (Karvonen formula)
    MHR = 217 - (0.85)A
    MHR = 206.3 - (0.711)A
    MHR = 205.8 - (0.685)A
    [ A=age years ]
  • Training Zone Heart Rate
    Upper End = (0.85)MHR + (.15)RHR
    Lower End = (0.6)MHR + (0.4)RHR
    [ RHR = resting heart rate in beats per minute ]

Why Do Some of These Formulas Contradict Each Other?

The physiologists who devised these mathematical models worked with different assumptions about the forms of the equations and the number of variables needed, and came up with the equation parameters using different samples of the human population. Therefore, two different equations for the same metric will not always agree, and may give vastly different answers when the input variables have extreme values -- e.g., very low or high ages, very low or high weights.


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