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.
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?
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 !
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! = -eiπ
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:
plus three times the square root of four
divided by seven
plus five times eleven
is nine squared and not a bit more