Friday, August 12, 2011

Pi: A magical number (and easier to say than "Phi")

The number Pi is a mathematical constant. It represents the ratio of the circumference of a circle to its diameter. If you divide the circumference of any circle (or perfectly-formed, hand-tossed pizza) by its diameter (the measure of a straight line cutting the pizza in half), you will get Pi, which is equal to 22/7, or approximately 3.14. Now let's review Fabian's quandary, from a couple of days ago...

Background:

Fabian Focaccia (not his real name, which is Sal Monella)  is a struggling 'artist' who is working at a neighborhood pizzeria in Brooklyn, New York (not his real location, as he is in the Federal Witness Protection Program, but which is still the best geographical location to make a pizza purchase if you/ youse should ever get around to it) to pay his bills until he can sell one of his paintings. He is faced with a decision and needs your help. He has cardboard boxes for 'take out' pizza (this pizza parlor does a big take-out business -- even for Arizona...oops!) which are each three inches deep (irrelevant for solving this problem) and measure exactly 20 inches by 20 inches square.

The Pizza Box Puzzle (In Two Parts):

1) What is the circumference of the largest pizza (assume that it is hand-tossed and perfectly round) which can be placed in the box neatly, i.e., placing it flat without stuffing it in and distorting its perfect shape?

Answer: This is really just a circle inscribed in a square. Here's a picture:

As you can see, the diameter of the pizza must  be exactly 20" for it to fit snugly in the box. If that's the case, then the pizza would have a circumference equal to Pi times the diameter, or approximately 62.8".

and,

2) What is the circumference of the largest individual pie (out of two) which can be placed in the box if two pies of equal size are placed in a cardboard container of the same dimensions as in number 1, above?

Answer: This is very similar to the first question. But this time you've got to put two equally-sized pies in the box without mutilating them. Here's a picture:

The two pies, side by side, can still not have combined diameters greater than the 20" length of the box (some of you were going to try doing this using the hypotenuse obtained by cutting the box into two right triangles, but that doesn't work - ha!). Each of the two pies will have a diameter of 10", and each will have a corresponding circumference of 31.4".

BCNU soon.

Douglas E Castle