Science · Middle school

The square-cube law

Scale a model up k times: its surface grows k², its weight k³. At what size do the legs break, and why are elephant legs so thick?

In the learning section below the figure, predict, try, complete the tasks and check yourself with short questions.

Fig. 1The square-cube law

Change k: the surface grows as k², the weight as k³; watch the stress on the gauge

Gauge (load ÷ strength): 0.1

Scale kk = 1

Leg thickness

Your own values

k (scale)

Safety margin

Safety margin: how many times the load they carry the legs of the k = 1 model can take before breaking (an assumption of the model).

k from 0.1 to 1,000, safety margin 1–1,000. A value applies when you press Enter or leave the box.

Scaled up

The legs hold

Length (k): ×1
Surface (k²): ×1
Volume and weight (k³): ×1
Leg cross-section (k²): ×1
Stress in the legs (k): ×1
Surface ÷ volume (1/k): ×1

Body: 72 unit squares on the surface, 36 unit cubes inside.

Gauge (load ÷ strength): 0.1 · With this rule the legs reach their breaking limit at k = 10.

Leg length ÷ diameter: ×1

Length, surface and volume (logarithmic axes)

length (k)
surface (k²)
volume (k³)

Learn

Progress is kept on this page only; no points are given.

1Predict
2Try
3Explain

If the model is made twice as big in every direction (length, width and height), how many times heavier does it get?

When a body is scaled up k times in every direction without changing its shape, its surface grows k² times and its volume and weight k³ times. Legs carry the weight on their cross-section, and a cross-section is a surface (k²). So the stress grows k³ ÷ k² = k times, and in a big enough model the legs break. The elephant rule is a thought experiment: if the leg diameter grows as k^1.5, the cross-section grows as k³ and the stress stays the same. In real mammals bone diameter grows roughly as M^0.36 (M: body mass); big animals mostly cope with upright posture. Model: one material, an animal standing still, the weight shared equally by four legs; the safety margin is an assumption. Sources: Galileo, Two New Sciences (1638), second day; J. B. S. Haldane, “On Being the Right Size” (1926); A. A. Biewener, Science 245 (1989).