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.
Archimedes
Greatest mind 250 BC
Saturday, January 29, 2011
Friday, January 28, 2011
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)
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)
Achimedes 250 BC The Greatest Mathematician Before Christ
Wow i get goose bumps just thinking about Archimedes. He is one of my favorite characters of all time. Thank goodness that by boys on Mythbusters (cable TV show) on Discovery channel have been highlighting Archimedes on several shows.
Let me first start off by saying that the Romans were trying to conquer Syracuse, Sicily (the "football" island at the end of the Italian BOOT !!...that is the only way students remember Italy...lol...or from the movie Patton...nah, they never saw Patton, that is an OLD movie)
Archimedes invented a few weapons. One of them was the gadipole. It would sling huge boulders at the Romans as they approach the castle. The Romans were aid to have said "Archimedes stood on top of the castle wall and lifted boulders weighing 500 lbs over his shoulders."
Another story goes that Archimedes was able to burn the Roman ships at sea as they crossed from Italy to Sicily. He used mirrors or some sort of glass and the sun's rays to set the ships on fire. The Roman soldiers said "Fire came from Archimedes EYES !!" He is a WITCH, a war lord, a magician, a Merlin, a GOD. (The boys at Myth Busters with the help of some MIT students tried this experiment and could NOT get it to work. Archimedes probably used he gadipole again to throw huge FIRE balls at the ships in the harbor.)
It is said that Archimedes was quoted as saying, "give me a place to put my lever and i will lift up the WORLD". And i believe him too.
So where do we start with the greatest mathematical mind Before Christ. (250 BC)
We have to mention Pythagoras 700 BC (another Greek who lived in Italy) and his famous formula a^2 + b^2 = c^2 (remember the 3-4-5 triangle ? i will have a blog on that soon)
and of course Euclid (300 BC) who lived in Alexandria, Egypt is probably the best known Greek mathematician. But he was just a book writer ("publish or perish" as the saying in the university goes). Alexander the Great conquered the know world (in Western Civilization history). And he brought all the great mathematical works to the University of Alexandria. Here Euclid was he department chair and wrote the number two best seller of ALL time...."The Elements". Which is still studied today in high school...."Euclidean Geometry" (i will write a blog about that too)
Getting back to Archimedes. The story goes that the king wanted to know if his crown was really made out of gold or not. So he asked Archimedes. Now Archimedes knew that one needed to know the Volume of the crown to know the density, but the crown was nice like a box (Volume = length X height X width).
While Archimedes was taking a bath, the idea came to him...."water displacement".
You put the crown in a tank of water and whatever amount of water that is displaced will give you the volume of the crown.
It is said that Archimedes was so excited that he ran down the street towards the King naked.....yelling "Eureka !!" (I found it !!)
COMMENTS:
(checked it out...interesting stuff...congrats!...from a 60 yr old lawyer)
Let me first start off by saying that the Romans were trying to conquer Syracuse, Sicily (the "football" island at the end of the Italian BOOT !!...that is the only way students remember Italy...lol...or from the movie Patton...nah, they never saw Patton, that is an OLD movie)
Archimedes invented a few weapons. One of them was the gadipole. It would sling huge boulders at the Romans as they approach the castle. The Romans were aid to have said "Archimedes stood on top of the castle wall and lifted boulders weighing 500 lbs over his shoulders."
Another story goes that Archimedes was able to burn the Roman ships at sea as they crossed from Italy to Sicily. He used mirrors or some sort of glass and the sun's rays to set the ships on fire. The Roman soldiers said "Fire came from Archimedes EYES !!" He is a WITCH, a war lord, a magician, a Merlin, a GOD. (The boys at Myth Busters with the help of some MIT students tried this experiment and could NOT get it to work. Archimedes probably used he gadipole again to throw huge FIRE balls at the ships in the harbor.)
It is said that Archimedes was quoted as saying, "give me a place to put my lever and i will lift up the WORLD". And i believe him too.
So where do we start with the greatest mathematical mind Before Christ. (250 BC)
We have to mention Pythagoras 700 BC (another Greek who lived in Italy) and his famous formula a^2 + b^2 = c^2 (remember the 3-4-5 triangle ? i will have a blog on that soon)
and of course Euclid (300 BC) who lived in Alexandria, Egypt is probably the best known Greek mathematician. But he was just a book writer ("publish or perish" as the saying in the university goes). Alexander the Great conquered the know world (in Western Civilization history). And he brought all the great mathematical works to the University of Alexandria. Here Euclid was he department chair and wrote the number two best seller of ALL time...."The Elements". Which is still studied today in high school...."Euclidean Geometry" (i will write a blog about that too)
Getting back to Archimedes. The story goes that the king wanted to know if his crown was really made out of gold or not. So he asked Archimedes. Now Archimedes knew that one needed to know the Volume of the crown to know the density, but the crown was nice like a box (Volume = length X height X width).
While Archimedes was taking a bath, the idea came to him...."water displacement".
You put the crown in a tank of water and whatever amount of water that is displaced will give you the volume of the crown.
It is said that Archimedes was so excited that he ran down the street towards the King naked.....yelling "Eureka !!" (I found it !!)
COMMENTS:
(checked it out...interesting stuff...congrats!...from a 60 yr old lawyer)
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