The Master Formulas

Dimensional Analysis

The dimensions of BMI are $[M L^{-2}]$. In SI units, this is $kg \cdot m^{-2}$. It is important to note that BMI is not a measure of density ($[M L^{-3}]$), though it is often confused with it. It is a Surface Density equivalent for a 3D object.

Variations: The "New BMI"

Some mathematicians argue that the standard formula penalizes tall people. The "New BMI" formula uses a $2.5$ power:$$New BMI = 1.3 \times \frac{Mass(kg)}{Height(m)^{2.5}}$$While not the medical standard, it is an excellent Calculus exercise to compare the rate of change ($d(BMI)/dh$) between the two models.

Shortcuts & Mnemonics

• The "Rule of 2": For a quick estimate, a height of $2m$ (very tall) means your BMI is roughly $1/4$ of your weight.
• Decimal Shift: Always square the height first. If your height is $1.x$, the square will be between $2.5$ and $4.0$ for most humans.
• Mnemonic: "Mass over Height-Square, keep it in the Air." (Reminding you that height is the denominator).

Edge Cases

• Athletes: High muscle mass (high $W$) leads to a high BMI even with low body fat. This is a False Positive in the model.
• The Elderly: Bone density loss can lower BMI while body fat percentage remains high (False Negative).
• Infants: BMI is not used for children under 2; instead, "Weight-for-length" percentiles are used, which follow a different growth curve.