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 = ( x d

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.

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

[pic]

The perimeter of square 2 is then 4 x a or [pic] 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.

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.

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

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.

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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