Body Mass Index (BMI) Formula
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Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| $BMI$ | Body Mass Index | kg/m² | A dimensionless index relating weight to height squared. |
| $w$ | Body weight | kg | Total body mass in kilograms. |
| $h$ | Height | m | Standing height in metres. |
What Is BMI?
Body Mass Index (BMI) is the most widely used screening tool to classify an adult's weight status relative to their height. It is calculated by dividing body weight in kilograms by the square of height in metres:
$$BMI = \frac{w_{kg}}{h_m^2}$$
A person who weighs 70 kg and stands 1.75 m tall has a BMI of 70 ÷ (1.75)² ≈ 22.9 kg/m², which falls squarely in the "normal weight" range.
Why Square the Height?
Squaring height accounts for the three-dimensional nature of the human body. If height doubles, surface area quadruples and volume (and thus mass) increases eightfold. The square approximates the relationship between height and body volume better than a simple ratio, making the index relatively comparable across people of different statures.
WHO Classification
| BMI Range | Category |
|---|---|
| < 18.5 | Underweight |
| 18.5 – 24.9 | Normal weight |
| 25.0 – 29.9 | Overweight |
| ≥ 30.0 | Obese |
The World Health Organization uses these cut-offs globally, though several Asian health authorities apply lower thresholds (e.g., overweight ≥ 23, obese ≥ 27.5) because Asian populations tend to carry more metabolic risk at lower BMI values.
What BMI Does Not Measure
BMI is a population-level screening tool, not a direct measure of body fat. It cannot distinguish between muscle mass and fat mass, nor does it reveal fat distribution. A heavily muscled athlete may register as "overweight" despite excellent metabolic health, while an elderly person with low muscle mass ("normal" BMI) may carry excess fat.
Common Misconceptions
- BMI predicts individual health — It does not. It identifies population risk, but individual assessment requires additional tests (waist circumference, body-fat percentage, blood markers).
- A lower BMI is always better — Underweight (BMI < 18.5) is associated with increased mortality risk from malnutrition, osteoporosis, and immune dysfunction.
- BMI is the same formula in imperial units — No. Imperial BMI requires a conversion factor of 703 (see BMI Imperial formula).
Historical Note
The formula was derived by Belgian mathematician Adolphe Quetelet in 1832 as a statistical tool to describe the "average man." It was popularised for clinical use by physiologist Ancel Keys in 1972, who coined the term "Body Mass Index."
Derivation & History
Adolphe Quetelet (1796–1874) observed that in populations of average build, body weight scaled roughly with the square of height rather than the cube. He published this "Quetelet Index" in 1832 as a statistical descriptor of human body proportion — not a clinical tool. For over a century it remained a curiosity of population statistics.
In 1972, American physiologist Ancel Keys analysed 7,400 men across five countries and found that weight/height² correlated best with direct measures of body fat among several candidate indices. He renamed it "Body Mass Index" and recommended it as the preferred anthropometric measure for epidemiological studies. The WHO subsequently adopted BMI as its standard weight-status classifier in the 1990s, cementing its global clinical use.
Worked Examples
Adult male, metric units
- Square the height: 1.78² = 3.1684 m²
- Divide weight by height squared: 82 ÷ 3.1684 ≈ 25.88 kg/m²
Result: BMI ≈ 25.9 — Overweight (WHO classification)
Adult female, metric units
- Square the height: 1.62² = 2.6244 m²
- Divide weight by height squared: 58 ÷ 2.6244 ≈ 22.10 kg/m²
Result: BMI ≈ 22.1 — Normal weight
Edge Cases & Limitations
Children and adolescents: BMI must be interpreted using age- and sex-specific percentile charts; adult cut-offs do not apply.
Pregnancy: BMI calculated before conception is used; weight gain during pregnancy is expected and normal.
Very short or very tall individuals: The formula systematically overestimates adiposity in short people and underestimates it in tall people, because body mass does not scale perfectly with height squared across extreme stature ranges.
Athletes with high muscle mass: Muscular individuals may be classified as overweight or obese despite low body-fat percentage.
Elderly: Loss of muscle (sarcopenia) may produce a "normal" BMI while hiding excess fat, underscoring BMI's limitation as a sole health indicator.
Real-World Applications
BMI is used in clinical screening, insurance underwriting, and public-health surveillance. Physicians use it as a first-pass triage to decide whether to order further metabolic tests. Epidemiologists track population-level BMI trends to measure the burden of obesity. Airlines and amusement parks use weight-for-height criteria (related to BMI) for safety calculations. Pharmaceutical clinical trials often use BMI as an inclusion/exclusion criterion. National health surveys (e.g., NHANES in the USA, KNHANES in Korea) report BMI distributions annually to guide policy.