Sphere

Sometimes you are brought up short by a revelation about something you've known for 50 years.  In the course of my very expensive education, I learned, and embedded, stuff at the age of 10 or 11 which the current generation of students are being taught at the age of 18. Take Latin: I was utter crap at Latin in school; I never did the home work and when required to 'construe' [translate] the doings of Caesar or Pompey would blurt out complete nonsense. But some of it stuck and it helps making sense of species names in biology. I won't easily forget how to calculate the area of a triangle [h * b/2] or the beautiful derivation [R] for that rote-learned formula.

In QM [remedial math] class last week, the kids were required to calculate the surface area of a solid hemisphere having been given the radius and the formula. It's not difficult but requires attention to check that all the elements have been accounted for before lashing down the answer spat out by the omni-present calculator.  The two elements are the area of a the flat circular base + the area of half of a sphere with the same diameter.  π*r² + 2π*r².  Last week, I was explaining to one of the lads that a solid hemisphere requires not only half the sphere but also the base [a very Latin construction that - non solum . . .  sed etiam was a cliché that appeared regularly in Caesar's Gallic Wars].  I am often brought up all standing by the holes in the education of the young: you can assume nothing as 'obvious to all thinking people'.  I final digression: in the pre-Quiz I set in the first class was "If the hub-caps on my car are 35cm in diameter [I measured, it's a Yaris], what is their surface area?" about 20% of the kids didn't know what was a hub-cap so their answer A = 35 x 35 = 1225 sq.com was correct in their hub-capless universe.

When you write π*r² + 2π*r² it becomes 'obvious' that the area of spherical part is exactly twice that of the flat base.  I may well have noticed that in my math-laden teenage years but it struck me as being almost mystically surprising and internally consistent last week.  It's obvious that you require more paint for the curve than the flat base but exactly 2x as much?

The fact that a sphere's surface area is 4π*r² is more difficult to prove than the triangle, so I'll refer you to two quite different methods on youtube: mostly geometry and mostly calculus.  Another observation known to Pythagoras and his pals was that the surface area of a sphere is exactly equivalent to a circumscribing cylinder [L] excluding the lids on the cylinder. The cylinder area is easy to work out if you can imagine getting some tin-snips [pause to deal with "Excuse me Dr Scientist, what are tin-snips?"] and cutting up the seam [think tin of beans] of the cylinder and flattening it out into a rectangle.  The shorter side or height of this rectangle [shown L] is 2r and the longer side is πd or π2r so the area is 2r * π2r = 4πr² !

I'm sorry if you think this is all obvious to the point of dull but it's qualitatively the sort of wonder that later and greater mathematicians reserved for the extraordinary connexion between π [the ratio between a circle and its diameter], e [the base of natural logarithms]  and i [the square root of -1] known as Euler's identity [formula R]. Gauss [the little snot] from his seat on Cloud Nine of Mathematics said "if this formula was not immediately obvious, the reader will never be a first-class mathematician".  It leads to a thigh-slappin' math-geek joke:
Q. How many mathematicians does it take to change a lightbulb?
A. One! = -e^iπ
And while we're on nerd-geek hilarity, I'll share a limerick that was going the rounds when my long-dead dad was in school and has recently surfaced to delight a smart young medical student of my acquaintance:

a dozen, a gross and a score
plus three times the square root of four
divided by seven
plus five times eleven
is nine squared and not a bit more