Archimedes’ crocodile

Our previous column stopped halfway through a great mathematical story. We wrote of Archimedes’ Eureka moments and his beautiful derivation of the volume of a sphere. We concluded with a promise to follow up with his equally beautiful derivation of the surface area of a sphere.

We’ll do that, but we first want to introduce you to Archimedes’ crocodile. Actually, we don’t know who found him originally, but the crocodile did eventually become Archimedes’ pet.

We’ve previously discussed a version of the crocodile. Take a pizza and chop it into an even number of equal slices.

Then, rearrange the slices into the big toothy smile of a crocodile.

How big is this smile? If the pizza has been sliced into a large number of thin pieces then the smile is very close to being a rectangle. The height of the “rectangle” will be close to the radius R of the pizza, and the base will be half its circumference.

We can then see at a glance that

Area of the pizza = Area of the smile = 1/2 x (Circumference of the circle) x R.

A slightly different method of obtaining this equation is to recognise that the area of each (roughly) triangular tooth has area 1/2 x base x R. Adding the bases, they sum to the circumference, and so the total area of all the teeth is 1/2 x (Circumference of the circle) x R.

As a consequence of this beautiful equation, if you know the area of a circle of radius R then you can determine its circumference, and vice versa. At which point, the familiar π enters the story.

What exactly is π? It’s actually a little tricky, but the easiest approach is to think of it as the precise number that makes true the following famous equation:

Area of a circle = πR²

(The trickiness is in knowing that the exact same number will work for every circle). By approximating circles by polygons, and by other methods, we can then obtain approximations to the size of the number π: the familiar 22/7 and 3.14, and so on.

Now, Archimedes’ crocodile has a message for us: since we have an exact formula for the area of a circle, we also have an exact formula for the circumference of a circle: rearranging the above smile equation, we find that

Circumference of a circle = 2πR

Waving goodbye to Archimedes’ crocodile, we now return to pondering spheres. As we wrote last week, Archimedes knew that the volume of a sphere of radius R is 4/3 πR³. But then Archimedes realised that the exact same pizza-crocodile trick works to determine the surface area of the sphere.

We simply have to think of the sphere as consisting of a very large number of thin pyramidal teeth. As we mentioned last week, each tooth will have a volume of 1/3 x (area of the base) x R. And, summing the bases of all the teeth gives the surface area of the sphere. So, summing the volume of the teeth,

Volume of sphere = 1/3 x (sum of the bases) x R = 1/3 x (Surface area of sphere) x R

Rearranging, we find, as Archimedes did,

Surface area of a sphere = 4πR²

Our final, crocodile-inspired, Eureka moment.

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator. 

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

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