Learn algebraic-geometric patterns for how changes in size affects surface area, volume and mass.
Be able to do numerical calculations for such changes
See how real world materials (e.g. bone) stand up to changes in scale.
Giant monster movies – Kaijū (怪獣 ) Japanese for “strange beast” – are a fun genre of cinema.
Godzilla, King Kong, Pacific Rim and Cloverfield. We see beasts of incredible size and mass
Live-action giant robot film is known as Tokusatsu (特撮)
How does changing the size of a creature affect it’s surface area, it’s volume, and it’s mass?
What real-world limiting factors will appear? These are questions of scaling.
The following is adapted from, Fran Bagenal, Astrophysical and Planetary Sciences, University of Colorado – http://lasp.colorado.edu/~bagenal/MATH/AreaVolume.html
Proportionality and the Value of Ratios
Consider a square – length “D” on a side:
The area is D x D = D2 (with the appropriate units – meters2 or km2 )
Now double the length of each side (each side is now 2D)
Area = 2D x 2D = (2D) 2 = 4 x D2 (with same units).
The ratio of sizes of the squares
= (Size Big)/(Size Small) = (2D)/(D) = 2.
The ratio of the areas of the squares
= (Area Big)/(Area Small) = (2D)2 / D2 = 22 = 4.
Next , let’s look at volume. Consider a volume of a cube that is D on a side.
Volume of cube = D x D x D = D3 (units are meters3 or km3 or feet3 )
Next consider a cube that is twice the size – 2D on a side. T
Volume = (2D)3 = 23 D3 = 8 D3.
Taking ratios we can summarize :
(Size Big)/(Size Small) = (2D)/ D = 2 -> We double the size
(Area Big)/(Area Small) = (2D)2/ D2 = 22 = 2 x 2 = 4 = [(Size Big)/(Size Small)]2
-> We get four times the surface area.
(Volume Big)/ (Volume Small) = (2D)3/ D3 = 23 = 2 x 2 x 2 = 8 = [(Size Big)/(Size Small)]3
-> We get 8 times the volume.
– – – – – –
What about cells, or planets and stars? They are spheres. Let’s start with circles: Begin with a circle of radius R.
Now double the size of the circle – double the radius (and/or double the diameter)
Area = p (2R)2 = = p ·22 R2 = π· 4 R2
Now move from 2 dimensions to 3 dimensions: from circles to spheres.
Area of a sphere = 4p R2
Now double the size (double length of the radius)
Area = 4p (2R)2 = 4p 22R2 = 16 pR2
Volume (of smaller sphere) = 4/3 pR3
Volume (larger sphere) = 4/3 p(2R)3 = 4/3 p23R3 = 4/3 x 8 pR3= 32/3 pR3
Taking ratios we get
(Area Big Circle)/ (Area Small Circle) = (p (2R)2) / ( p R2) = 22= 4
(Area Big Sphere)/ (Area Small Sphere) = (4p (2R)2) / ( 4p R2) = 22= 4
(Volume Big Sphere)/ (Volume Small Sphere) = (4/3 p (2R)3) / ( 4/3 p R3) = 23= 8
What about irregular shapes? Will the math be more complex – or easy What’s the formula for the area of a frog?
Area of a frog = “something” x H2
Now, what happens if the frog is twice as big?
Area of big frog = “something” x (2H)2= “something” x 22H2 = “something” x 4 H2
Volume of a frog = “something else” x H3
Volume of the big frog
= “something else” x (2H)3
= “something else” x 23H3
= “something else” x 8 H3
We have no idea what the “something else” is.
But guess what? Doesn’t matter. Take ratios!
(Area Big Frog)/ (Area Small Frog) = (something x (2H)2) / (something x H2) = 22= 4
(Volume Big Frog)/ (Volume Small Frog) = (something else (2H)3) / (something else H3) = 23= 8
Conclusion? All you need to think about is the ratio.
Ratio of areas is always (Area Big)/ (Area Small) = [(Size Big)]/(Size Small)]2.
Ratio of volumes is always (Volume Big)/(Volume Small) = [(Size Big)]/(Size Small)]3.
Area is proportional to Size2
Volume is proportional to Size3
This works for squares, spheres, frogs,gorillas (King Kong!), dinosaurs (Godzilla), etc.
Godzilla vs Scaling Laws of Physics
Adapted from “Godzilla Versus Scaling Laws of Physics,”
Thomas R. Tretter, University of Louisville, Louisville, KY
The Physics Teacher Vol.43, Nov. 2005
By the 17th century the importance of scaling effects on structural strength was noted by Galileo (Dialogues Concerning Two New Sciences, 1638)
A horse falling eight feet would break many bones
A dog falling eight feet would walk away, usually with no broken bones.
A dog could carry two or three similarly sized dogs on its back.
A horse could not carry even one similarly sized horse.
Godzilla, 1998, with Matthew Broderick.
View scene at 10:15 – footprint
The scale of the image is estimated from the size of a case that Broderick is carrying: the case is approximately the same length as his back, from waist to shoulder.
On screen (measured with ruler):
case length = 2 cm
foot length = 28 cm, foot width = 23 cm
Real length of a person’s back = about 52 cm
case length = about 52 cm
Therefore 1 cm on screen = 26 cm in real life.
Length of foot = 26 * 28 cm = 728 cm
Width of foot = 26 * 23 cm = 598 cm
Approximating her feet as rectangular,
area of Godzilla’s 2 feet = (728 cm)(598 cm)(2) = 870,688 cm2
– – – – –
Bone strength is proportional to cross section area of the bone
Major source of stress on the leg bones: the weight being supported.
Let’s find the stress pressure exerted on our own leg bones.
Measure the circumference of an ankle, then find the cross sectional area of your two ankles together.
Ankle circumference C = 25 cm = 2πr
r = 25 cm / (2π) = 4.0 cm
Area = πr2 = 50 cm2
For two ankles, area = 100 cm2
Average person’s weight (presses down on ankles) = 165 lb
Convert to metric units:
1 pound = 4.45 newtons -> 165 lb = 734 N
Pressure = Force / area = 734 N /100 cm2 = 7.3 N/cm2
Now let’s do this for an elephant: Foot circumference = 52 in (130 cm)
Weight = 9300 lb = 4.1 x 10^4 N
Results in 7.5 N/cm2 , pressure exerted by each foot.
Pressures are similar across species regardless of size.
Conclusion? Pressures are related to the compressive strength of bone material.
They can’t be greatly exceeded regardless of how large the animal may be.
26 minutes: Godzilla stomps NYC
Estimate the area of Godzilla’s ankles, as compared to the area of her feet
Ankle is, at best, about 1/4 the area of the foot
Using previous data, estimate 2.2 x 10^5 cm2 for total cross sectional area of the two ankles.
Estimate Godzilla’s weight.
34:00 Post rampage scenes including a building with a huge hole through it
Size of this hole is used to estimate that Godzilla’s volume is 8.5 x 10^4 m3 (see Fig. 2).
Assume that Godzilla’s density is approximately that of water, 1000 kg/m3, results in a weight of 8.3 x 10^8 N.
This value together with the ankles’ cross sectional area gives a pressure of about 3.8 x 10^3 N/cm2!
This is about 500 times greater pressure on the bone than for other creatures.
2016 Massachusetts Science and Technology/Engineering Curriculum Framework
Appendix VIII Value of Crosscutting Concepts and Nature of Science in Curricula
In grades 9–12, students can observe patterns in systems at different scales and cite patterns as empirical evidence for causality in supporting their explanations of phenomena. They recognize that classifications or explanations used at one scale may not be useful or need revision using a different scale, thus requiring improved investigations and experiments. They use mathematical representations to identify certain patterns and analyze patterns of performance in order to re-engineer and improve a designed system.
Next Gen Science Standards HS-PS2 Motion and Stability
Crosscutting Concepts: Different patterns may be observed at each of the scales at which a system is studied and can provide evidence for causality in explanations of phenomena. (HS-PS2-4)
A Framework for K-12 Science Education
Scale, proportion, and quantity. In considering phenomena, it is critical to recognize what is relevant at different measures of size, time, and energy and to recognize how changes in scale, proportion, or quantity affect a system’s structure or performance….
The understanding of relative magnitude is only a starting point. As noted in Benchmarks for Science Literacy, “The large idea is that the way in which things work may change with scale. Different aspects of nature change at different rates with changes in scale, and so the relationships among them change, too” . Appropriate understanding of scale relationships is critical as well to engineering—no structure could be conceived, much less constructed, without the engineer’s precise sense of scale.
- Dimension 2, Crosscutting Concepts, A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012)