Formulas Reference

Scientific references for all calculators used in the fitness-nutrition skill.

BMI (Body Mass Index)

Formula: BMI = weight (kg) / height (m)²

Category	BMI Range
Underweight	< 18.5
Normal	18.5 – 24.9
Overweight	25.0 – 29.9
Obese	30.0+

Limitation: BMI does not distinguish muscle from fat. A muscular person can have a high BMI while being lean. Use body fat % for a better picture.

Reference: Quetelet, A. (1832). Keys et al., Int J Obes (1972).

TDEE (Total Daily Energy Expenditure)

Uses the Mifflin-St Jeor equation — the most accurate BMR predictor for the general population according to the ADA (2005).

BMR formulas:

• Male: BMR = 10 × weight(kg) + 6.25 × height(cm) − 5 × age + 5
• Female: BMR = 10 × weight(kg) + 6.25 × height(cm) − 5 × age − 161

Activity multipliers:

Level	Description	Multiplier
1	Sedentary (desk job)	1.200
2	Lightly active (1-3 days/wk)	1.375
3	Moderately active (3-5 days)	1.550
4	Very active (6-7 days)	1.725
5	Extremely active (2x/day)	1.900

Reference: Mifflin et al., Am J Clin Nutr 51, 241-247 (1990).

One-Rep Max (1RM)

Three validated formulas. Average of all three is most reliable.

• Epley: 1RM = w × (1 + r/30)
• Brzycki: 1RM = w × 36 / (37 − r)
• Lombardi: 1RM = w × r^0.1

All formulas are most accurate for r ≤ 10. Above 10 reps, error increases.

Reference: LeSuer et al., J Strength Cond Res 11(4), 211-213 (1997).

Macro Splits

Recommended splits based on goal:

Goal	Protein	Fat	Carbs	Calorie Offset
Fat loss	40%	30%	30%	−500 kcal
Maintenance	30%	30%	40%	0
Lean bulk	30%	25%	45%	+400 kcal

Protein targets for muscle growth: 1.6–2.2 g/kg body weight per day. Minimum fat intake: 0.5 g/kg to support hormone production.

Conversion: Protein = 4 kcal/g, Fat = 9 kcal/g, Carbs = 4 kcal/g.

Reference: Morton et al., Br J Sports Med 52, 376–384 (2018).

Body Fat % (US Navy Method)

Male:

BF% = 86.010 × log₁₀(waist − neck) − 70.041 × log₁₀(height) + 36.76

Female:

BF% = 163.205 × log₁₀(waist + hip − neck) − 97.684 × log₁₀(height) − 78.387

All measurements in centimeters.

Category	Male	Female
Essential	2-5%	10-13%
Athletic	6-13%	14-20%
Fitness	14-17%	21-24%
Average	18-24%	25-31%
Obese	25%+	32%+

Accuracy: ±3-5% compared to DEXA. Measure at the navel (waist), at the Adam's apple (neck), and widest point (hip, females only).

Reference: Hodgdon & Beckett, Naval Health Research Center (1984).

APIs

• wger: https://wger.de/api/v2/ — AGPL-3.0, exercise data is CC-BY-SA 3.0
• USDA FoodData Central: https://api.nal.usda.gov/fdc/v1/ — public domain (CC0 1.0)