Archimedes - Indirect Reasoning

Some things seem so obvious to us today, but back in 300 or 250 BC, they were not invented yet.

On of those things was how to find the area of a DONUT (a cirlce inside another circle sharing the same center...called "concentric circles" or the donut).

What is the area between this two circles ? or what is the area of the donut.

I know most of you already know how to do this but Archimedes had to use something called "indirect reasoning". In other words the same way that we can not find the volume of the crown in the first post BELOW, Archimedes had to figure a way to find the area indirectly.

It is "quite elementary my boy" like Sherlock Holmes would say.

The answer is: to find the area of the BIG one and then subtract the area of the small one. Simple right ?

Lets say the radius of the big circle is 10 and the radius of the small circle is 5.

The area of the big circle would be A = pi (10)^2 = 100 pi.

and the area of the small circle is A = pi (5)^2 = 25 pi.

Therefore the area of the donut is 100 pi - 25 pi = 75 pi !! found indirectly.

But now we have to have a discussion of pi (I am sure there is a way of getting a Greek letter on this computer...where is my son when i need him ?...lol)

the Greeks going back all the way to the beginning 800 BC, knew that if you take the Circumference of a circle (the perimeter....the distance around the outside of a circle) and divided by the diameter (the distance from one side to the other going through the center of the circle) you would always get 3.14.

C/d = 3.14

Even in the old testament of the Bible, it says that it took 21 steps around the Temple and only 7 steps across the temple..which comes out to 21 / 7 = 3 !! (close enough). And it does not matter whether it is a temple or a tree or an Oreo cookie (that is what we use in class when we do this exercise), it will always come out 3.14

In honor of the Greeks, we call this number 3.14 = pi (a greek letter, the first letter in the greek word for perimeter, "περίμετρος" ).

π (sometimes written pi) is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in the Euclidean plane.

How did Archimedes approximate the value of PI ?

Well first he drew a circle with diameter of 1. The Circumference would then be
C = pi * diameter (remember C/diam = pi...so when we multiply both sides by diameter, we get C = pi * d).

Therefore a circle with diameter one, has a circumference of PI !!

Now Archimedes put a square around the circle. (go ahead get your paper and pencil)

What is the perimeter of the square around this circle ? It should be 4 !!

so is the circumference of the circle more or less than the perimeter of the square around it ? YES, the circumference has to be LESS, therefore PI is less than 4 !!

pi < 4

Now Archimedes put a hexagon inside the circle. (A Hexagon has 6 sides. The pentagon in Washington, DC has 5 sides.)

With a little of Euclidean geometry we can determine the hexagon inside the circle to have edges of length (1/2). And (1/2) times 6 sides is equal to 3 !!

The perimeter of the hexagon inside the circle has a perimeter of 3 !!

The circle is on the outside of the hexagon, so the circumference of the circle has to be GREATER than 3, and therefore pi must be greater than 3 !!

SO now we know that 3 < pi < 4 !!

That was a very good start for Archimedes in 250 BC.

Now the genius of Archimedes comes through. Instead of a square, he uses an eight sided polygon...the STOP sign...and then 16-sided polygon...and then 32 sides.

He discovers that 3.14 < pi < 3.15 is the value of pi !!

(now remember Archimedes did NOT have decimals...they had not been invented yet...lol.
He of course used fractions, but you get the idea)

This by the way is called the "Squeeze Theorem". (if you do not know the value of something, you just squeeze it between two values that you do know and get a good approximation)