The area of a circle is one of the first formulas that you learn as a young math student. It is simply taught as, . There is no explanation as to why the area of a circle is this arbitrary formula. As it turns out the area of a circle is not an easy task to figure out by your self.
Early mathematicians knew that area was, in general to four sided polygons, length times width. But a circle was different, it could not be simply divided into length and width for it had no sides. As it turns out, finding the measurement to be squared was not difficult as it was the radius of the circle. There was another aspect of the circle though that has led one of the greatest mathematical voyages ever launched, the search of Pi.
One of the first ever documented estimates for the area of a circle was found in Egypt on a paper known as the Rhin d Papyrus around the time of 1650 BCE. The paper itself was a copy of an older “book” written between 2000 and 1800 BCE and some of the information contained in that writing might have been handed down by Imhotep, the man who supervised the building of the pyramids. The paper, copied by the scribe named Ahmes, has 84 problems on it and their solutions. On the paper, in problem number 50 he wrote; “Cut off 1/9 of a diameter and construct a square upon the remainder; this has the same area as a circle.” Given that we already know that the area of a circle is we find that the early Egyptian estimate for the area of a circle was which simplified to or 3. 16049… Though, the papyrus does not go into detail as to how Ahmes derived this estimate.
This estimate for Pi given by the ancient Egyptians is less than 1% off of the true value of Pi. Given, there was no standard of measurement in that day and they also had no tools to aid them in such calculations such as compasses or measuring tapes, this is an amazingly accurate value for Pi and the area of a circle. Another early attempt at the area of a circle is found in the Bible. In the old testament within the book of Kings Vii. 23 and also in Chronicles iv. 2 a statement is made that says; “And he made a molten sea, ten cubits from one brim to the other; it was round all about and his height was five cubits: and a line of thirty cubits did compass it round about.” From this verse, we come to the conclusion that Pi is 30/10 or simply 3.
The book of Kings was edited around the time of 550 BCE. Much better estimates were already at hand in the day and much earlier, though they must not have been known to the editors of the Bible. The Babylonians also played an early hand at the area of a circle but it wasn’t known until 1936 when a Babylonian tablet was unearthed. It states that a ratio of the perimeter of a hexagon to the circumference of a circumscribed circle equals in modern terms (the Babylonians used a numerical system that was base 60 and not base 10 as we use today). One of the reasons they chose the hexagon was because the perimeter of a hexagon is exactly equal to six times the radius of the circumscribed circle.
This is. Since the definition of Pi is the circumference divided by Diameter, we come to. Therefore, the equation in turn gives us or. This is just under the true value of Pi.
Most early estimates for Pi were no more exact than saying that Pi was greater than but less than. Most of the methods for solving for the area of a Circle are also unknown as to how they were derived. Many scholars deduce that early estimators of the circle were able to find their measurements by a ways of rearranging. For example, if you have a rectangle, and you cut off a triangle from one end of the rectangle then reattach it to the opposite side in which the triangle came from, then you now have a parallelogram of equal area to which the rectangle had before. Applying this type of thinking to a circle, we must first cut a circle into four equal parts. Placing the parts side by side so that their flat side lay against each other and the round outer edges face the outside.
You continue to do this process to smaller and smaller sections of a given circle. Given that you will reach an infinite amount of pieces that are rearranged in this way you will be left with a rectangle that has one side that is length and another side that is width. The resulting area of this arranged rectangle is, the area of a circle. This method is later discovered on a Japanese paper roll dated to 1693, which was later used by Leonardo Da Vinci. Another great mind in the history of the area of a Circle was Archimedes of Syracuse. Noted for his naked run through Syracuse shouting “Eureka!” after having solved a problem while taking a bath Archimedes derived a new way in which to find the area of a circle.
Consider a circle of radius 1 which is circumscribed by a polygon of sides, with semi perimeter bn. Another polygon of sides, with semi perimeter an, super scribes the circle such as the chart on the next page demonstrates. The diagram below demonstrates the case n = 2, with the hexagons having 6 sides. The goal of this procedure is to make it such that. Through this infinite series the polygons converge upon the circle and form a circle that overlaps the original circle.
Though, Archimedes didn’t have Calculus to aid his search. Through only geometrical means, Archimedes determined that Pi lay somewhere between and. So, let = K sin (? /K) and = K tan (/K). If we assume that K is the number of sides that the polygon has, namely sides, then, through mathematical induction, and. Archimedes used this convention to arrive at a polygon with 96 sides, or n = 6, which led him to Pi being in between and which is. Let it be noted that Archimedes did this estimation without Trigonometry, Calculus, and decimal notation.
This method of finding the area of a Circle was used until recent times where computers have taken over the main task of finding the never ending search for all the number s in Pi. There are many more very good estimates for the area of a circle throughout history. Though, at the time at which Archimedes made his discovery of a truly accurate way to find the true vale of Pi, there was a split in thought. One side followed Archimedes and took his formula to the side to find Pi to over 35 accurate decimals, by hand. The other side followed the past to a less productive area of mathematics. These people have come to be known as the “Circle Squarers.” Much like the early attempts of rearranging the circle into a Rectangle, these people try and try again to fit the area of a circle into that of a square using only a compass, straight edge, and a pencil.
Many a great mathematicians wandered over to this once in their career and usually came to the conclusion that Pi = , which is a common solution to the problem among circle squarers. It wasn’t until 1882 when Ferdinand Lindemann proved that Pi was a transcendental number that the claim of the Circle Squarers was finally thrown out. Pi, being a transcendental number, cannot be expressed as a finite algebraic equation and thus, cannot be reconstructed as a square through Euclidean Geometry. Though, even after Lindemann’s historical find, Circle Squarers, now called cyclometers, continued to pour their incorrect theories onto the mathematical world.
Most of them state that one great mathematician was wrong, or that there is a conspiracy hiding the true value of Pi. The problem with Cyclometers is that they can show you they are right, however cannot prove it using conventional mathematics. For instance, a cyclometer can show you a circle of diameter one and roll it along a ruler and show you that the circumference of the circle is exactly 3. 1415. Some even dare to argue that great mathematicians such as Pythagoras and Archimedes were incorrect, the rest of the mathematical world doesn’t dare question their founding mathematicians, and that they alone, the cyclometer, have discovered the true value of Pi. One circle squarer even went so far as to submit a law in his home state of Indiana that his value of Pi be used as the legal value of Pi.
It was passed, but to this day awaits further legislation in regard to its factuality. In the end, there is still an ongoing search for the true area of a Circle in continued research of the number Pi. Scientists today have reached a record number of decimals of Pi to 206, 158, 430, 000 using a Hitachi Supercomputer. The calculation took 37 hours, 21 minutes and 4 seconds. Using the latest calculation for Pi, if you were to assemble a circle a million miles in diameter, the circle would be less than an inch off. But why the pursuit of a solution that will never end? For many, being that there are no perfect circles even in nature, the perfect circle is an unattainable goal to seek.
Through the adventure of discovering new aspects about the circle, other insights may be revealed. The mystery of the circle is an endless pursuit, but for mathematicians, it is the pursuit of perfection.