Why Bigger Doesn't Mean Proportionally More: The Scaling Math Behind Pet Food and Aquarium Stocking
2026-09-14
It feels obvious: a bigger dog needs proportionally more food, and a bigger tank can hold proportionally more fish. Twice the size, twice the amount — simple ratio math. Except it isn't true in either case, and the reason is the same underlying idea in both: size doesn't scale in one dimension, but the thing you care about often only scales with one dimension of it.
The 0.75 power law hiding in your dog's food bowl
The Dog Food Calculator on this site computes a Resting Energy Requirement (RER) as 70 × (body weight in kg)^0.75 — not simply 70 × weight. That exponent, 0.75 instead of 1, is Kleiber's law: across mammals, metabolic rate scales slower than body mass. A smaller animal has more surface area relative to its volume, loses heat faster, and burns more energy per kilogram just to stay warm and running. A larger animal is relatively more efficient per kilogram.
Put numbers on it: a 10 kg dog has an RER of 70 × 10^0.75 ≈ 393.6 kcal/day. Double the weight to 20 kg, and RER becomes 70 × 20^0.75 ≈ 662.5 kcal/day — a 1.68× increase, not 2×. If you "did the obvious thing" and just fed the bigger dog exactly double the smaller dog's portion, you'd be overfeeding it by close to 20%, every single day. Life-stage multipliers (puppy, neutered adult, active adult, senior) stack on top of RER to get the full Daily Energy Requirement, but the 0.75 exponent is the part that breaks the "just scale it linearly" instinct before life stage even enters the picture.
Why a "long" tank holds more fish than a "tall" tank of the same volume
The Aquarium Stocking Calculator runs two independent rules side by side on purpose, and they can disagree — which is the point. The volume-based rule is a metric, size-tiered version of the old "inch of fish per gallon" guideline. The surface-area rule instead looks only at the tank's footprint (length × width), because gas exchange — oxygen dissolving in, CO2 escaping — happens at the water's surface, not throughout the water column. A tank's height barely matters for how much oxygen the water can pick up; its footprint matters a great deal.
Two tanks can both be labeled "20 gallons" and still have very different real capacity. A tall, narrow 20-gallon tank (roughly 12″ × 12″ × 16″) has a footprint of about 144 in². A long, low 20-gallon tank (roughly 30″ × 12″ × 12″) has a footprint of about 360 in² — two and a half times more surface area, for identical volume. If you stocked both tanks by volume alone, you'd be treating them as equivalent when, for oxygen availability, they aren't close.
The common thread: don't collapse a shape into a single number
Both calculators resist the temptation to reduce "how big is it" to one number and multiply. A dog's energy need depends on weight raised to a fractional power, not weight itself. A tank's stocking capacity depends on its footprint, not its volume. In both cases, the honest calculation needs more than the one number most people would reach for first — which is exactly why both calculators ask for the actual inputs (weight and life stage; length, width, and height) instead of a single "size" field.
- Doubling a dog's weight increases its energy need by roughly 1.68×, not 2× — Kleiber's law, the same 0.75-power relationship seen across mammals.
- Two tanks with identical volume can have very different real stocking capacity if their footprints differ — height barely affects oxygen exchange, footprint does.
- When something is described by a single "size" number, check whether the thing you actually care about scales with that number linearly, or with only part of it.