What is the allometric equation?

Y = a × M^b, where Y is the biological trait, M is body mass in kg, a is the normalization coefficient (Y when M = 1 kg), and b is the scaling exponent. When b = 1 the trait is directly proportional to mass (isometric); b 1 means it grows faster.

What is Kleiber's law?

Kleiber's law states that basal metabolic rate scales with body mass to the 3/4 power (b ≈ 0.75). Proposed by Max Kleiber in 1932, it holds across more than 20 orders of magnitude in body mass and is one of the most robust empirical regularities in biology.

Why do scaling exponents cluster near multiples of 1/4?

The quarter-power family (b = 1/4, 1/2, 3/4, −1/4 …) is thought to arise from the fractal-like geometry of biological distribution networks such as the cardiovascular system. The West-Brown-Enquist model derives these exponents from first principles of network optimization.

Are the predictions exact?

No. Allometric equations are statistical fits across many species. Individual animals deviate due to phylogeny, diet, activity level, and environment. Use the tool for order-of-magnitude estimates and comparative analysis, not precise physiological measurements. It is not intended for preclinical or clinical drug dose conversion, which requires compound-specific pharmacokinetics and BSA/Km normalization.

Body mass is converted to kilograms internally. Preset coefficients are calibrated for kilograms, so outputs are in the standard unit for each parameter: watts for BMR, bpm for heart rate, years for lifespan, grams for organ masses, and so on.

Can I use this for non-mammalian species?

Most presets are fitted to mammalian data. Birds and reptiles often share similar exponents but different a values. Use Custom mode and enter taxon-specific values from the literature for non-mammalian groups.

Can I use this calculator to scale drug doses across species?

No. Pharmacological dose extrapolation (such as determining a Human Equivalent Dose from animal models) requires compound-specific pharmacokinetic parameters, clearance rates, and body surface area normalization (such as FDA Km factors). This calculator models general biological traits and is not intended for pharmacological dosing.

Allometric Scaling Calculator

Enter body mass and choose a biological parameter to compute its scaled value using the allometric equation Y = a × M^b.

Allometric Scaling Inputs

Biological Parameter

Body Mass

Coefficients are calibrated for kilograms; other units are converted automatically.

Comparison Mass (optional)

Compare a second animal side by side (same unit as above).

Results

Primary mass result

— —

—

Y = a × M^b

Enter values and click Calculate.

Coefficient a —

Exponent b —

Scaling type —

Scaling Curve (log-log)

Summary

Allometric scaling describes how biological traits change with body size using the power-law equation Y = a × M^b, where M is body mass, a is the normalization coefficient, and b is the scaling exponent. This tool provides preset exponents for common parameters — basal metabolic rate (Kleiber's law, b ≈ 0.75), heart rate (b ≈ −0.25), lifespan (b ≈ 0.25), brain mass, organ masses, and more — as well as a custom mode for any coefficient and exponent you specify.

How it works

Select a biological parameter from the preset list, or choose "Custom" to enter your own coefficient and exponent.
Enter body mass in kilograms (or switch to grams, pounds, or metric tons via the unit selector).
Optionally enter a second comparison mass to see both values side by side with a ratio.
Click "Calculate" — the tool applies Y = a × M^b and displays the result with units.
A log-log chart shows how the parameter scales across the full mammalian mass range.
Read the formula breakdown to see the exact numeric substitution used.

Use cases

Estimate basal metabolic rate for any mammal from body mass using Kleiber's law.
Compare trait ratios between small and large organisms using side-by-side mass inputs.
Compare predicted organ masses against measured values from literature.
Estimate resting heart rate or lifespan for a hypothetical animal.
Teach quarter-power scaling laws in physiology or ecology courses.
Extrapolate home range or population density across species of different sizes.
Validate experimental measurements against allometric predictions.
Explore how exponents differ between taxa using the custom mode.

Frequently Asked Questions

Last updated: 2026-09-30 · Reviewed by Nham Vu