Calculator of Alexandria

BMI calculator

Calculate body mass index from weight and height in metric or imperial units, and see where the formula and its categories come from.

Solve for

Adult categories (WHO): under 18.5 underweight, 18.5–24.9 healthy, 25–29.9 overweight, 30 and over obesity.

Formula

BMI=mh2\text{BMI} = \frac{m}{h^2}
How it's derived

The formula

BMI=mh2\text{BMI} = \frac{m}{h^2}

mm is weight in kilograms and hh height in metres, giving BMI in kg/m².

Where it comes from

BMI isn’t derived from physiology. It’s an empirical index, and its history explains its shape.

Why divide by height at all? Taller people weigh more, so raw weight can’t compare people of different heights. We want a ratio that stays roughly constant across heights for people of similar build.

Why squared, not cubed? If people scaled up uniformly in every direction, weight would grow with volume, as h3h^3. In the 1830s the Belgian statistician Adolphe Quetelet measured real adults and found that weight grows more slowly than that. Taller adults aren’t proportionally wider or deeper. The best simple fit to his data was close to h2h^2, so the ratio

mh2\frac{m}{h^2}

stays roughly level across heights. Ancel Keys named it the body mass index in 1972, after showing it tracked body fat better than other simple weight–height ratios.

The categories. The WHO cut-offs (18.5, 25 and 30) are round numbers chosen where large population studies show health risks starting to rise. They are thresholds by convention, not natural boundaries.

Imperial units. With pounds and inches you need a conversion factor: BMI=703×lb/in2\text{BMI} = 703 \times \text{lb}/\text{in}^2, because 0.4536 kg/lb÷0.02542 m2/in2≈7030.4536\ \text{kg/lb} \div 0.0254^2\ \text{m}^2/\text{in}^2 \approx 703.

Worked example

70 kg and 175 cm:

BMI=701.752=703.0625=22.9\text{BMI} = \frac{70}{1.75^2} = \frac{70}{3.0625} = 22.9

Watch out for

  • BMI can’t tell muscle from fat. Athletes often read as overweight, and older adults with low muscle mass can read as healthy.
  • The adult categories don’t apply to children, who are assessed with age- and sex-specific percentiles.
  • It’s a screening number, not a diagnosis.

Sources

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