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How Archimedes found the area of a circle

"Squaring the Circle"

The problem of determining the area of a circle was once considered a great mathematicalchallenge. On this page, we will discuss a method that Archimedes (287-212 BCE) used forsolving this problem. (This problem was called "squaring the circle": i.e. trying to find the squarethat has the same enclosed area as a circle of a given radius.)

Ofcourse, we now know that

The whole point will be to actually show that this is true, by somehow computing the area of thecircle from known areas of simpler shapes.

A reminder about Pi

The irrational number is defined as the ratio of the circumference of a circle to its diameter,that is:

We will use this result in the final steps of the method below.

Step 1: Approximating the circle with a square

The diagram here shows a square inscribed in a circle. (Inscribed means that it exactly fits inside,with its vertices just touching the edge of the circle). Let us look at the geometry more closely.

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We notice that the diagonals of the square are the same as thediameter of the circle, and so have length . Thus the sides of the square, AB, BC which are equal ofcourse, must have length

. (Since ABC is a right triangle, whose hypotenuse is AC).Thus the area of the inscribed square is thus

This is still not a very good approximation to the true area of the circle, but it is a start.

Step 2: Approximating the circle with a hexagon

If we use a polygon with more sides to try to approximate the area of the circle, we would hopeto get a better result. So consider an approximation which uses the hexagon, as shown in thediagram.

To find the area of the hexagon, we might subdivide it into sixtriangles, whose area is easily computed if we know the heightand the base of any one of the (all equal) triangles.

One such triangle is shown in the diagram at left: Observe that itssides AC and AB are of length , since they are radii of thecircle. Further, notice that the angle ABC should be 60 degrees(i.e. one sixth of a revolution, 360/6 =60 degrees). This impliesthat the sum of the other angles, CAB plus CBA is 120 degrees(since the sum of all the angles in a triangle is 180 degrees).Further, since they are equal (opposite equal sides), they, too,

must be 60 degrees. So we have, hopefully, convinced you thatthe triangles in this hexagon are all equilateral: all three sides areof length . We leave it up to you to further establish that the

height of each of these triangles is , so that the area of thehexagon, it follows, is:

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This is closer to the true area of the circle, but still not very close.

Step n: Using an n-sided polygon

Suppose we increase the number of sides in the polygon and lookat the general case, in which there are n sides. Then the area of then sided polygon will be n times the area of one of the triangles, i.e.

where are, respectively the base and height of one of thetriangles shown in this picture. Now note what happens as thenumber of sides, n increases:

In this expression, the term is the perimeter of the polygon, which, as n increases, becomescloser and closer to the circumference of the circle. Further, the height of the triangle,approaches the radius of the circle, so that, as we approximate the circle by a polygon with more

sides, i.e. a greater number of (thinner) triangles, we find that the area approaches:

We have shown above, that the area of the circle does indeed involve that special constantwhich arose as the ratio of the circumference to the diameter of the circle.

By dragging the red ball in the diagram below, you can experiment with the number of sides inthe polygon used to approximate the area of the circle. You should notice that as n increases, thearea of the polygon becomes a closer and closer fit to the area of the circle.

Drag the red ball. As the number of sides, n, in the polygon increases, its area becomes a

closer and closer approximation to the area of the circle.

For your consideration:

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  (1) Show that the length of each side of the square in the first step we did is .  (2) Show that the height of the triangles that form the hexagon, in the second step

we discussed is .  (3) Suppose you were to consider a 36-sided polygon. In that case the angle of the

tip of each of the triangular subdivisions would be 360/36 = 10 degrees. This isnot a "standard" angle, but we could certainly use trigonometry to establish thelength of the base and the height of such a triangle. What would you get for thearea of this 36-sided polygon? How does this area compare with the true area of the circle?

This is the last time we will be using triangles to compute areas for a little while. From now on,you will see that rectangular subdivisions will form a much more important unit of subdivision in

computing areas of irregular regions in the plane.However, the concept of increasing the number of units to better approximate some complicatedshape will be very basic. It will appear and reappear in many examples, as it forms one of thekey ideas in integral calculus.

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Mathematicians and scientists have always been intrigued by pi, but it acquired a whole newfollowing when it foiled a diabolic computer in a Star Trek episode. wears different hats -- it isthe ratio of a circle's circumference to the diameter, it is a transcendental number (a number thatcannot be the solution of an algebraic equation with integral coefficients).

One of the most curious methods for computing is attributed to the 18th century Frenchnaturalist, Count Buffon and his Needle Problem. A plane surface is ruled by paralled lines, all dunits apart. A needle of length at least d is dropped on the ruled surface. If the needle lands on aline, the toss is considered favorable. Buffon's amazing discovery was that the ratio of favorabletosses to unfavorable was an expression involving pi. If the needle's length is equal to d units, theprobability of a favorable toss is 2/pi. The more tosses, the more closely did the resultapproximate. In yet another probability method to compute pi, R. Charles, in 1904 found theprobability of 2 numbers (written at random) being relatively prime to be 6/pi.

It's startling to discover the versatility of pi, crossing as it does the wide spectrum of geometry,calculus and probability.

Buffon's Needle

Buffon's Needle 

The Buffon's Needleproblem is amathematical method of 

approximating the valueof pi

involving repeatedlydropping needles on asheet of lined paper andobserving how often theneedle intersects a line.

Buffon's Needle

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The method was first used to approximate π by Georges-Louis Leclerc, the Comte de Buffon, in1777. Buffon was a mathematician, and he wondered about the probability that a needle wouldlie across a line between two wooden strips on his floor. To test his question, he apparently threwbread sticks across his shoulder and counted when they crossed a line.

Calculating the probability of an intersection for the Buffon's Needle problem was the firstsolution to a problem of geometric probability. The solution can be used to design a method forapproximating the number π.

Subsequent mathematicians have used this method with needles instead of bread sticks, or withcomputer simulations. We will show that when the distance between the lines is equal the lengthof the needle, an approximation of π can be calculated using the equation

Monte Carlo Methods

The Buffon's needle problem was the first recorded use of a Monte Carlo method. Thesemethods employ repeated random sampling to approximate a probability, instead of computingthe probability directly. Monte Carlo calculations are especially useful when the nature of theproblem makes a direct calculation impossible or unfeasible, and they have become morecommon as the introduction of computers makes randomization and conducting a large numberof trials less laborious.

π is an irrational number, which means that its value cannot be expressed exactly as a fractiona/b, where a and b are integers. This also means that a normal decimal repetition of pi is non-

terminating and non-repeating, and mathematicians have been challenged with trying todetermine increasingly accurate approximations. The timeline below shows the improvements inapproximating π throughout history. In the past 50 years especially, improvements in computer 

capability allow more decimal places to be determined. Nonetheless, better methods of approximation are still desired.

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 A recent study conducted the Buffon's Needle experiment to approximate π using computer software. The researchers administered 30 trials for each number of drops, and averaged theirestimates for π. They noted the improvement in accuracy as more trials were conducted.

These results show that the Buffon's Needle method approximation is relatively tedious.Compared to other computation techniques, Buffon's method is impractical because the estimatesconverge towards π rather slowly. Even when a large number of needles were dropped, this

experiment gave a value of π that was inaccurate in the third decimal place.

Regardless of the impracticality of the Buffon's Needle method, the historical significance of theproblem as a Monte Carlo method means that it continues to be widely recognized.

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Lucas number

The Lucas numbers are an integer sequence named after the mathematician François ÉdouardAnatole Lucas (1842 – 1891), who studied both that sequence and the closely related Fibonacci

numbers. Lucas numbers and Fibonacci numbers form complementary instances of  Lucassequences. 

Definition

Like the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediateprevious terms, i.e. it is a Fibonacci integer sequence. Consequently, the ratio between twoconsecutive Lucas numbers converges to the golden ratio. However, the first two Lucas numbersare  L0 = 2 and  L1 = 1 instead of 0 and 1, and the properties of Lucas numbers are thereforesomewhat different from those of Fibonacci numbers.

A Lucas number may thus be defined as follows:

The sequence of Lucas numbers begins:

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ... (sequence A000032 in OEIS) 

Extension to negative integers

Using Ln-2 = Ln - Ln-1, one can extend the Lucas numbers to negative integers to obtain a doublyinfinite sequence :

..., -11, 7, -4, 3, -1, 2, 1, 3, 4, 7, 11, ... (terms Ln for are shown).

The formula for terms with negative indices in this sequence is

Relationship to Fibonacci numbers

The Lucas numbers are related to the Fibonacci numbers by the identities

 

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  , and thus as approaches +∞, the ratio approaches

  

Their closed formula is given as:

where is the Golden ratio. Alternatively,  Ln is the closest integer to .

Congruence relation

Ln is congruent to 1 mod n if n is prime, but some composite values of n also have this property.

Lucas primes

A Lucas prime is a Lucas number that is prime. The first few Lucas primes are

2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, ... (sequence A005479 in OEIS) 

If  Ln is prime then n is either 0, prime, or a power of 2.[1]  L2m is prime for m = 1, 2, 3, and 4 and

no other known values of m.

Lucas polynomials

The Lucas polynomials  Ln( x) are a polynomial sequence derived from the Lucas numbers in thesame way as Fibonacci polynomials are derived from the Fibonacci numbers. Lucas polynomialsare defined by the following recurrence relation:

Lucas polynomials can be expressed in terms of Lucas sequences as

The first few Lucas polynomials are:

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The Lucas numbers are recovered by evaluating the polynomials at  x = 1. The degree of  Ln( x) isn. The ordinary generating function for the sequence is