Body Mass Index (BMI) Calculator

Find what BMI does to a body that changes size without changing shape. This is the Reference adult: 175 cm and 70 kg.

1. Start with the index as defined. The numerator is a mass and the denominator is an area, so BMI is not a ratio of like quantities — and that mismatch is what the rest of the problem is about.

   $$\mathrm{BMI}=\frac{m}{h^2}=\frac{70}{1.75^2}=\frac{70}{3.0625}=22.857kg/{\mathrm{m}}^2$$

2. Now enlarge the whole body: multiply every length by s and hold the density fixed. Mass follows volume and gains s³ while the denominator gains only s², so one power of s survives the division.

   $$h\to sh,m\to s^{3}m⟹\mathrm{BMI}\to \frac{s^{3}m}{(sh{)}^2}=s\cdot \mathrm{BMI}$$

3. An index that survives the scaling has to match the exponent of mass, and cubing the height does exactly that. The ponderal index is then identical for every member of the family — not approximately, exactly.

   $$\mathrm{PI}=\frac{m}{h^3}=\frac{70}{5.359375}=13.061kg/{\mathrm{m}}^3,\mathrm{PI}\to \frac{s^{3}m}{(sh{)}^3}=\mathrm{PI}$$

4. Writing BMI as the ponderal index times height makes the height dependence explicit, so no mass needs recomputing at each size. 4.6 BMI units separate the 155 cm and the 190 cm version of one body, with not a gram of extra fat anywhere in it.

   $$\mathrm{BMI}(h)=\mathrm{PI}\cdot h:h=1.55\Rightarrow 20.24,h=1.75\Rightarrow 22.857,h=1.90\Rightarrow 24.82$$

5. The category boundaries are fixed numbers while the index is not, so the crossing heights are a single division.

   $$\mathrm{PI}\cdot h=25\Rightarrow h=\frac{25}{13.061}=1.914\mathrm{m};\mathrm{PI}\cdot h=18.5\Rightarrow h=1.416\mathrm{m}$$