A new study reveals that sharks, regardless of size or shape, closely follow the two-thirds scaling law, a centuries-old mathematical rule, offering fresh insight into evolution, development, and how animals adapt to their environments.

From tiny lantern to massive whale, sharks come in a stunning variety of shapes and sizes! From hand-sized deep-sea lantern sharks that glow in the dark to 20-metre-long whale sharks gliding through tropical waters, they are one of nature’s most diverse groups. But despite their differences, all sharks face the same biological challenge: how to efficiently move oxygen, heat, and nutrients through their bodies.

Now, a new scientific study has revealed that sharks, no matter how big or small, follow a centuries-old mathematical rule called the two-thirds scaling law, which helps explain how evolution shapes the bodies of animals.

Published in Royal Society Open Science, the research shows that shark body shape changes with size in a very predictable way, matching the theory nearly perfectly.

What is the two-thirds scaling law?

The rule is based on a simple idea in geometry:

  • Surface area increases with the square of length.
  • Volume increases with the cube of length.

This means that as an animal gets bigger, its surface area doesn't grow as fast as its volume. For living organisms, this balance matters greatly because:

  • Surface area affects how efficiently an animal can take in oxygen, release heat, and absorb nutrients.
  • Volume determines how much energy and oxygen the body needs to stay alive.

So, the surface area-to-volume ratio becomes a key factor in how animals grow and function. Until now, this principle had mostly been tested only on small organisms, like insects or cells.

Why sharks were perfect for this study

Though it may seem unusual, sharks are ideal animals to test this mathematical rule because they come in a wide range of sizes, from 20 cm to over 20 metres, and have different shapes and lifestyles.

They include reef-dwelling species, deep-sea hunters, and hammerheads with unusual head shapes. Yet all of them must balance similar biological needs: getting oxygen from water through gills, maintaining body heat, and moving efficiently.

Also, sharks are ecologically important and under growing threat, so learning more about their biology has value beyond academic interest.

How scientists measured sharks in 3D

To study this, researchers created detailed 3D models of 54 shark species using high-tech methods like CT scans and photogrammetry, a technique that uses photos to build a 3D structure.

They refined these models using Blender, a popular 3D design tool, and then measured the surface area and volume of each shark.

By applying a method called phylogenetic regression, which accounts for shared ancestry among species, the team tested whether shark shapes follow the two-thirds law.

What the results showed

The findings of the were remarkable:

Sharks’ surface area scales with their volume raised to the power of 0.64, which is just 3% off from the predicted value of 0.67 (or two-thirds).

This close match means that sharks, from smallest to largest, evolve in a way that sticks closely to this ancient mathematical rule.

One reason may be developmental constraints, natural limits on how animals can grow. Changing the surface-area-to-volume ratio too much could mean rewiring how the body forms during embryonic development, something evolution generally avoids.

The importance of the study

This discovery isn’t just about sharks. The two-thirds law is used in many biological and climate science models that calculate how animals regulate heat, consume oxygen, and respond to environmental stress.

Until now, most of these models lacked real-world data from large animals like sharks. With this new research, scientists can now build more accurate predictions, not just for sharks, but possibly for whales, reptiles, and other large creatures.

At a time when the climate is changing fast and biodiversity is under pressure, understanding how animals function at different sizes helps us protect them more effectively.