Download archimedes squares the circle

ARCHIMEDES SQUARES THE CIRCLE
Jacques Chaurette
May 31, 2007
Archimedes (287 BC to 212 BC) was the smartest man of his time
and they now say has anticipated Newton with the invention of
calculus 1700 years earlier.
The greek letter pi  is used to represent the ratio of a circle’s
circumference to its diameter. The Babylonians knew the value of 
to be 3.125, not bad but not good enough for Archimedes. We now
know that pi is equal to 3.1419…
The relationship between the diameter d and circumference c of a
circle is:
c=xd
If we have a circle that has a 1 unit diameter and you can use
whatever unit you want: 1 foot, 1 meter, 1 kilometer, then the
circumference c of these various circles will be c = 3.14 ft, 3.14 m,
and 3.14 km. Using a unit value for d means that once we have
calculated the value of the circumference c we automatically know 
because when d =1, c = .
Archimedes’ brilliant idea was to square the circle. If you put a
square inside a circle of unit diameter, one of the sides of the square
will be proportional to the diameter in some fashion and once you
know the side of the square it’s easy to find its perimeter, just
multiply that value by 4. This will give a low value for . But wait a
minute, what if you put a square on the outside of the circle. One of
its sides will be equal to the diameter d, therefore we can easily
calculate its perimeter. And now we have a larger value for the
perimeter or circumference of the circle. The true value of c is
between the perimeter of the outer square and the inner square. The
values obtained for the two square perimeters are too far apart to give
a precise value for c. What Archimedes realized was if he turned his
squares into many sided polygons he could easily calculate the
perimeter of these polygons and if he added more sides he could come up with a very precise value
for the circumference of the circle c and therefore the value of , and that’s how he did it.
For those of you who want to see the details of this little exercise, follow me.
The perimeter of square 1 is equal to 4 x d, and since d is fixed at 1 unit for our purpose then the
perimeter is equal to 4 units.
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To find the perimeter of square 2 we need the help of our old friend Pythagoras. The inner square
forms a right angle triangle with the hypotenuse equal to d and the other two sides are unknown but
equal since we have a square.
From Pythagoras we know that d2 = a2 + a2, and since d = 1 then 1 = a2 + a2 therefore
1 = 2 x a2 or
a
The perimeter of square 2 is then 4 x a or
4
2
1
2
which is equal to 2.82
So with this simple exercise we now know that the value of c or pi is between 2.82 and 4. That’s a
good start but we are not even as close as the Babylonians, we need to add more sides to our
squares.
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Let’s go to a pentagon, which is a five-sided polyhedron.
Now we have to find the perimeter of a pentagon, Pythagoras won’t help us here. Pythagoras is
useful when we know two sides of a right-angle triangle and we are looking for the third. We are
looking for the length of one of the sides of pentagon 2, we will call half of one side a which we
will determine with the diameter d. Unfortunately we only have the length of one side of the
triangle to determine a and that’s not enough, but we have the angle between the edges of the
pentagon. Since the pentagon has five sides, the angle between each side is 360/2 = 75 degrees, or
for half a side it will be 36 degrees.
Here’s where we have to introduce a little trigonometry. With trigonometry we can figure out the
length of a side of a triangle knowing the length of another side and one of the angles that is not 90
degrees.
The value of a is d/2 x sin 36
It’s very easy to find the value of sin 36 many calculators have this function. The value is 0.587.
Now you might say Archimedes didn’t have a calculator, how could he determine the value of sin
36. Well he was a crafty fella and he could do it but I think this is worth spending a bit more time
on it’s own and for now please accept that the value of the sine of an angle is easily calculated and
this helps us calculate the length of the side of a triangle knowing the length of one side and an
angle.
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The value of a is then d/2 x 0.587 and since d = 1 then
a = ½ x 0.587 = 0.2935
The perimeter of pentagon 1 is the length of the side which is a x 2 times the number of sides or
Perimeter of pentagon 1 = a x 2 x 5 = 0.2935 x 2 x 5 = 2.935
We do a similar exercise to calculate the value of the perimeter of the outside pentagon, pentagon
1. In this case we are looking for length b, which is half the length of a side of pentagon 1.
In this case the value of b
b = d/2 x tan 36
tan means tangent and tan 36 has a value easily calculated by many calculators.
The value of tan 36 is 0.726. The value of b is then d/2 x 0.726 and since d = 1 then
b = ½ x 0.726 = 0.3632
The perimeter of pentagon 2 is the length of the side which is b x 2 time the number of sides or
Perimeter of pentagon 2 = b x 2 x 5 = 0.3632 x 2 x 5 = 3.632
Therefore the value of the circumference c or  is now between 2.935 and 3.632. We are getting
closer to the real value of 3.1419.
If we use a dodecahedron which has 12 sides
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The perimeter of the inside dodecahedron is 2 x 12 x ½ x sin (360/12/2) = 3.105
and
the perimeter of the outside dodecahedron is 2 x 12 x ½ x tan (360/12/2) = 3.215
The value of the circumference c or pi is now between 3.105 and 3.215, again closer to the real
value of 3.1419.
If we use a 25 sided polyhedron which has 12 sides
the perimeter of the inside dodecahedron is 2 x 25 x ½ x sin (360/25/2) = 3.133
and
the perimeter of the outside dodecahedron is 2 x 25 x ½ x tan (360/25/2) = 3.158
The value of the circumference c or pi is now between 3.133 and 3.158, again closer to the real
value of 3.1419.
We don’t need to go much further, if we took a polyhedron of 50 sides, we would be right on the
money or at least close enough for government work.
Today, we calculate the value of  to any precision that we desire. It’s done using a formula that’s
called a series. It’s another easy thing to understand but would be the topic of another article.
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Mathematicians classify the number  as a transcendental number. There
are many such numbers, one of them is the exponential e used as the base
of the natural logarithm. This class of numbers cannot be defined with a
final or ultimate precision, there would need to be an infinite amount of
numbers after the decimal point and that’s not possible. We can see the
reason based on the exercise we just did. We get a more and more precise
value for  the more sides we add to our polyhedron. But no matter how many sides we add we can
always pack another side in, that means we will get a more refined result with more decimals.
Another interesting thing about the number  is that all the
decimals are completely random; there are no sequences of
any kind that repeat. So it has been used on occasion to
generate random numbers. Why do you need random
numbers, you may ask? Well, there are many reasons but one
of them is to decide who amongst a group of people will be
chosen to do a certain experiment and in what order. This
avoids bias on the part of the experimenter and possibly
influencing the results.
The value of pi to 50 decimal places is:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
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