The BMI formula

The BMI formula has only two variables — your weight and your height — and three operations: a multiplication, a division and a square. The whole thing fits in a single line, but the imperial form (the one with the magic-looking number 703) hides a tidy little bit of unit-conversion algebra. This page walks through both forms, derives the 703 from first principles, and shows three worked examples you can verify by hand.

Written by Aftab. Aftab is an AI engineer, a fitness coach with six years of experience, a self-taught fitness enthusiast and the software developer who built this calculator as a hobby project. Health information on this page is sourced from the WHO, the CDC, the NHLBI, and peer-reviewed literature; the clinical content is also signed off by a certified professional listed on the clinical review network before publication. Learn more about how we verify our content.

The two-line summary

If you only remember two formulas, remember these:

• Metric: BMI = weight (kg) / height (m)²
• Imperial: BMI = 703 × weight (lb) / height (in)²

Both produce the same number for the same person. Our calculator accepts either unit and does the conversion for you.

Where the formula came from

The history is covered in detail on What is BMI? — the short version is that the Belgian polymath Adolphe Quetelet published the ratio in 1832 and the physiologist Ancel Keys popularised the name "Body Mass Index" in 1972 after showing it correlated with body fatness better than rival indices [1].

Why height squared?

A natural first guess would be that weight should scale linearly with height — if a 5-foot person should weigh x, a 6-foot person should weigh 6/5 × x = 1.2x. But biological scaling is not linear. Rubner's 1883 work on basal metabolic rate in mammals showed that energy use scales with body mass to the 2/3 power, which is the same scaling that surface area (∝ length²) shares. A taller person is geometrically similar to a shorter one, so a doubling of height should — at constant body shape — quadruple the cross-sectional area that supports the weight. Squaring the height in the denominator normalises for that, which is why the formula works as well as it does.

This is also the reason other scaling exponents (height³, the Ponderal Index, the Benn Index) have been proposed and studied — they can fit the data slightly better in specific populations but never replaced kg/m² because the simplicity mattered more than the marginal gain [2].

Where does the 703 come from?

The imperial form needs a conversion factor because the units of the ratio change when you swap kilograms and metres for pounds and inches. The factor falls out of dimensional analysis:

We want a number k such that BMI = k × lb / in² gives the same value as BMI = kg / m². Convert the imperial units to metric:

• 1 lb = 0.45359237 kg (exact, by the 1959 international yard-and-pound agreement)
• 1 in = 0.0254 m (exact)

So

BMI = (0.45359237 × lb) / (0.0254 × in)²
     = (0.45359237 / 0.0254²) × lb / in²
     = (0.45359237 / 0.00064516) × lb / in²
     = 703.069… × lb / in²

Rounded to a whole number that the public can remember, that's 703. (The NIH uses 703 in its patient-facing materials; some older texts use 704.7 from a different rounding chain. Both round-trip to within ±0.1 of the metric value, which is finer than any band boundary.)

Three worked examples

Example 1 — metric

A 70 kg adult, 1.75 m tall.

BMI = 70 / (1.75)² = 70 / 3.0625 = 22.86
Result: BMI 22.86 → "Normal weight" (18.5–24.9).

Example 2 — imperial

The same person, in pounds and inches: 70 kg = 154.32 lb, 1.75 m = 68.90 in.

BMI = 703 × 154.32 / (68.90)² = 108486.96 / 4747.21 = 22.85
Result: BMI 22.85 — the 0.01 difference is rounding in the conversion. Both forms agree.

Example 3 — child (CDC LMS)

A 10-year-old boy, 140 cm, 45 kg. The BMI number is the same formula:

BMI = 45 / (1.40)² = 45 / 1.96 = 22.96

But the interpretation is different — see the children's section of the BMI guide. The calculator on this site uses the CDC's 2022 extended LMS reference [3] to convert 22.96 at age 10 for a boy to a z-score of +1.75 → 96th percentile → "Obesity" band.

Why the units are not interchangeable

A common mistake is to write BMI = weight (lb) / height (in)² with no conversion factor, which produces a number about 14× too small (a BMI 22.86 person would "score" 0.0325 by that formula). Some online calculators and gym apps still get this wrong; if your result looks like a decimal, suspect a missing 703.

Likewise, some calculators use centimetres instead of metres in the denominator. If you use cm, the result is about 10,000× too small — divide by 10,000 to recover the correct number (or just use metres: 170 cm = 1.70 m).

Inverse problems

The formula is also useful in two "backwards" directions, both of which the calculator exposes:

1. Healthy weight for a given height. Solve weight = BMI × height² for the WHO healthy band (18.5–24.9). At 1.75 m, the range is 56.7–76.3 kg.
2. Target BMI for a given weight. Solve height = √(weight / BMI) for the same band.

FAQ

Why divide by height squared and not cubed?

The Ponderal Index (weight / height³) is an alternative proposed in 1921 by F. D. Rohrer. It slightly outperforms BMI in children and very tall adults. BMI won out historically because the simpler units (kg/m² vs kg/m³) made hand calculation easier before calculators were common, and the additional precision was not worth changing decades of clinical convention [2].

Is there a BMI formula for children?

The BMI number itself uses the same formula at every age. The difference is in the interpretation: for ages 2–19 the CDC publishes percentiles by age and sex (see our explanation) rather than the fixed WHO adult cut-offs.

Why not use lean body mass instead?

Lean body mass is a more informative number but requires a DEXA scan, a Bod Pod, or a multi-frequency bio-impedance device — none of which are free, instant and available to everyone. BMI is the cheap, universal proxy. The two are complementary, not competitors; see BMI vs body fat percentage.

References

1. Keys A, Fidanza F, Karvonen MJ, Kimura N, Taylor HL. Indices of relative weight and obesity. J Chronic Dis. 1972;25(6):329–343. https://doi.org/10.1016/0021-9681(72)90027-6
2. Deurenberg P, Deurenberg-Yap M, Guricci S. Asians are different from Caucasians and from each other in their body mass index/body fat per cent relationship. Obes Rev. 2002;3(3):141–146. https://doi.org/10.1046/j.1467-789X.2002.00065.x
3. U.S. Centers for Disease Control and Prevention, National Center for Health Statistics. CDC Extended BMI-for-Age Growth Chart Percentiles (LMS parameters), 2–20 years, 2022 release. Bundled as Official docs/bmi-age-2022.csv. https://www.cdc.gov/growthcharts/extended-bmi.htm
4. National Heart, Lung, and Blood Institute, NIH. Calculate your body mass index. Patient information page, accessed June 2026. https://www.nhlbi.nih.gov/health/educational/lose_wt/BMI/bmi-m.htm

Last updated: 6 June 2026. No content on this page constitutes medical advice.

Next

Read the full category guide, see where BMI falls short, or try the calculator with your own numbers.