Archimedes - Area of a Circle

So now we have C = pi * diam and we know that pi = 3.14, but what is the area of this circle ?

In comes Archimedes. We knew that the area of the polygon is equal to the area of all the triangles that make up the polygon.

A funny story about the King of France. Every time a map maker made another copy of the map, the French Empire got bigger and bigger (like the map of Russia got bigger and bigger during the Cold War with the USA...yes, it was a distorted map of the world....as Russia got bigger, so did Greenland...lol).

Now mathematicians have been using triangles to measure the area of polygons for many centuries going back to the Greeks (and i do not mean the frats at the university kind of Greeks...that is for my younger son who just joined an "academic" frat in college...they must drink smart beer)

So the King commissioned the Royal Court mathematician to measure his kingdom. After 7 hard years of traveling the land with an army of surveyors, the mathematician presented his accurate map to the King. The Kingdom had SHRUNK by 50%. OFF WITH HIS HEAD !! was the reward for such good work.

Archimeded knew that the Area = (the area of one triangle)*(the number of sides)

Lets look at our Hexagon again. It is made up of 6 triangles. Each triangle is
A=(1/2)bh (by thw way only 13% of graduating high schoolers remember this formula and we have already lost 30% of the population that dropped out of high school because they had to work to support the family....i would hate to guess what percent of the adults know this formula for the area of a triangle...i did see it once on the side of gallon of paint). The best way to remember this formula forever is to remember that the area of a rectangle is A=bh or A=lw....and that the triangle occupies (1/2) that area. Do NOT memorize the formula, learn where it comes from and you will have a better chance of remembering it later in life. (of course this will not help you if you are lost in the woods and need to survive, but it might help if you plan to paint a house or wall.)

Anyways if we have a polygon of "n" sides, we say the Area = (1/2)bh * n

Now if you draw this polygon (lets say a hexagon) (ok, go get your paper and pencil)

You will see that if you multiply the bases of all those triangles times the number of sides "n" you will get the perimeter of the polygon.

So A = (1/2) ph....where p is perimeter and h is the distance from the center to each edge of the polygon.

Now comes the genius of Archimedes. He takes "n" to infinity !! (this is what Calculus is all about...going to infinity). As n goes to infinity, the polygon becomes more and more like a circle.

Now the question is what is happening to the perimeter ? As n goes to infinity, the Perimeter gets closer and closer to the Circumference of the circle. And C= pi * d.

Meanwhile the height of the triangles get closer and closer to the radius of the cirlce. So by substitution we get:

n -> to infinity, p -> to circumference, h -> to radius and polygon becomes a circle.

A = (1/2) C * r

But remember C = pi *d or C = 2 pi r

So now we have A =(1/2) (2 pi r) r

or A = pi r^2 !!!

the formula for the area of a circle...and we can now also find the area of the donut (the first post below) and we can have that donut with our coffee....

The genius of Archimedes. The Wizard.