Height for Weight: The Ideal Height–Weight Relation
Enter your weight to see the heights at which it falls in the World Health Organization’s normal BMI range, and the height for which the Broca, Hamwi, Devine, Robinson and Miller “ideal weight” formulas would call it ideal. For weight by height, use the Ideal Weight Calculator.
- Broca (height − 100)165.0 cm
- Hamwi (1964)174.4 cm
- Devine (1974)173.9 cm
- Robinson (1983)177.5 cm
- Miller (1983)174.6 cm
How it is calculated
This page is the inverse of the Ideal Weight Calculator: it takes weight instead of height. An adult cannot change their height, so the result is not a target but a reading of the height–weight relation: “for someone of which height would this weight be a normal BMI?” For health, the measure is the World Health Organization’s adult BMI classification, in which 18.5 to 24.99 kg/m² is normal. Broca’s rule (normal weight = height in cm − 100) is attributed to the French physician Paul Broca and dated to 1871; Laurent and colleagues’ 2020 review notes it was first used for “normal weight” and later for “ideal weight”. The “height − 105” or “10–15% less” versions often given for women differ between sources and no original basis for them could be found, so they are not used here. The Hamwi (1964), Devine (1974), Robinson (1983) and Miller (1983) formulas are conventions developed for drug dosing and clinical calculations and written for a base height of 5 ft (152.4 cm); as Pai and Paloucek (2000) review, they look alike because the height-weight tables behind them agree. WHO’s classification (TRS 894, Table 2.1) defines a normal range, not a single “optimal” BMI, so the tool gives no single target such as “BMI 22”.
Height range = √(weight ÷ BMI limit) · Broca height = weight + 100 · Formula height = 152.4 + (weight − base) ÷ coefficient × 2.54The BMI limits are WHO’s normal range (18.5–24.99 kg/m²). Base and coefficient (kg; male / female): Hamwi 106 lb + 6 lb / 100 lb + 5 lb (1 lb = 0.45359237 kg); Devine 50 + 2.3 / 45.5 + 2.3; Robinson 51.65 + 1.85 / 48.67 + 1.65; Miller 56.2 + 1.41 / 53.1 + 1.36 (per inch over 5 ft). Broca does not depend on sex.- Since BMI = weight ÷ height² (m), height = √(weight ÷ BMI). Dividing the weight by the upper limit and taking the square root gives the shortest height; by the lower limit, the tallest.
- The tool gives the shortest and tallest height, in 0.1 cm steps, at which this weight’s BMI to two decimals is in the normal range.
- Broca’s rule is “weight = height − 100”, so its inverse is height = weight + 100 cm.
- For the other formulas subtract the base from the weight, divide by the coefficient (inches over 5 ft), multiply by 2.54 and add 152.4 cm. If the result is below 152.4 cm it is not shown, as the formula is not defined there.
Example
For 65 kg, √(65 ÷ 24.99) = 1.6128 m and √(65 ÷ 18.5) = 1.8744 m; in 0.1 cm steps the normal range is 161.3–187.4 cm (at 161.2 cm the BMI is 25.01). The Broca height is 65 + 100 = 165 cm. For a woman, Devine: (65 − 45.5) ÷ 2.3 = 8.478 in, 152.4 + 8.478 × 2.54 = 173.9 cm.
Frequently asked questions
What should my ideal height be for my weight?
An adult’s height does not change, so an “ideal height” is not a target. The tool only shows the heights at which your weight is a WHO normal BMI (18.5–24.99 kg/m²). To judge your own weight, calculate your BMI with your real height and talk the result over with your doctor or a dietitian.
Is the “Brock formula” the same as Broca’s?
Yes; “Brock formula” is a misspelling of Broca’s rule. Sources attribute the rule to Paul Broca (1871); we found no source for the name “Paul Brock” or an 1883 date.
Why do some formulas give no result?
The Hamwi, Devine, Robinson and Miller formulas start from a base weight at 5 ft (152.4 cm) and are not defined below that height. If your weight is less than a formula’s base weight, the matching height is below 152.4 cm and the tool does not show it.
Can children use this tool?
No. WHO’s adult BMI cut-offs and the formulas are for adults; for ages 5 to 19, WHO reads BMI against its growth reference for age and sex.
Source
- WHO Technical Report Series 894 (2000): Obesity — preventing and managing the global epidemic, Table 2.1 (p. 9): normal 18.50–24.99 kg/m²
- Laurent I et al. The use of Broca index to assess cut-off points for overweight in adults: a short review. Rev Endocr Metab Disord 2020;21(4):521–6, abstract (Broca 1871, height − 100)
- Walter E, Patel A. Variability in ideal body weight formulae. J Intensive Care Soc 2023;24(3 Suppl):16–7 (Broca, Mémoires d’anthropologie, 1871)
- McCarron MM, Devine BJ. Case number 25: gentamicin therapy. Drug Intell Clin Pharm 1974;8(11):650–5 (Devine formula)
- Robinson JD et al. Determination of ideal body weight for drug dosage calculations. Am J Hosp Pharm 1983;40(6):1016–9, abstract (coefficients)
- Miller DR, Carlson JD, Lloyd BJ, Day BJ. Determining ideal body weight (and mass). Am J Hosp Pharm 1983;40(10):1622–5
- Peterson CM et al. Am J Clin Nutr 2016;103(5):1197–203, Table 3: Hamwi (1964), Devine, Robinson and Miller equations
- Pai MP, Paloucek FP. The origin of the “ideal” body weight equations. Ann Pharmacother 2000;34(9):1066–9 (review of origins)
- NIST SP 811, Appendix B.8: inch = 2.54 cm (exact), pound = 0.4535924 kg
- WHO, Growth reference data for 5–19 years (2007): BMI-for-age (5–19 years), BMI reference by age and sex