Prove the Circumference Formula
Introduction:
Archimedes is credited to be the creator of the circumference formula, but more importantly to find the first theoretical calculation of Pi. Pi is an irrational number and the digits are continuous and never come to an end. Archimedes knew he had only found an approximation of pi, he found that pi is between 3 1/7 and 3 10/71. “Pi is a name given to the ratio of the circumference of a circle to the diameter. That means, for any circle, you can divide the circumference (the distance around the circle) by the diameter and always get exactly the same number. It doesn't matter how big or small the circle is, Pi remains the same. Pi is often written using the symbol  and is pronounced "pie", just like the dessert”(math.com).
The circumference formula we use to this day is:

Pi is abbreviated to 3.14 when being written, but calculators use a much more accurate version of the number.
I chose to investigate this topic because the origin of formulas interests me. Somehow these letters are created and it works under any conditions and never seems to fail. Prior to this investigation I have learned what the circumference formula is and how to apply it.
Statement of Purpose:
The main purpose of this investigation is to prove the circumference formula to be correct. Through this investigation I will use different processes of math to prove this formula correct. This will show that the formula holds true in multiple settings.
Plan of Investigation:
I will derive the formula and work it in multiple ways to prove that the formula we have used for the past centuries to be correct. I will also look at the history of pi.
Math:

The above picture shows all the parts you need to know for this ex...
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...en surpassed until 1429, when astronomer Jamshid Al-Kashi of Samarkand found  Correct to sixteen decimal places. Western mathematicians did not surpass the Tsus approximation until around 1600.
Works Cited
“Circumference of a Circle - Derivation." Derivation of the Formula for the Circumference of a Circle. Math Open Refrence, n.d. Web. 28 Oct. 2013..
"PI." PI. Math.com, n.d. Web. 28 Oct. 2013. .
"Johann Heinrich Lambert." The world of π. N.p.. Web. 11 Nov 2013. .
"The "Jewish" or "Bible" Value of "pi"." Purplemath. Elizabeth Stapel, n.d. Web. 11 Nov 2013. http://www.purplemath.com/modules/bibleval.htm.
Howard, Eves. An Introduction to the History of Mathematcis. Fifth Edition. New York: The Saunders Series, 1983. Print.
This shows that there is a difference of 2cm between A and B, and B
Abstract—The transition to calculus was a remarkable period in the history of mathematics and witnessed great advancements in this field. The great minds of the 17th through the 19 Centuries worked rigorously on the theory and the application of calculus. One theory started another one, and details needed justifications. In turn, this started a new mathematical era developing the incredible field of calculus on the hands of the most intelligent people of ancient times. In this paper, we focus on an amazing mathematician who excelled in pure mathematics despite his physical inability of total blindness. This mathematician is Leonard Euler.
So using this formula but with the data we collected from our first attempt, this is what it would look like; Tan(60°) x 23m = 39m. As you can tell this answer collected from our first attempt is very well incorrect, but at the time, our group did not know this.
After 3rd century BC, Eratosthenes calculation about Earth's circumference was used correctly in different locations such as Alexandria and syene (Aswan now) by simple geometry and the shadows cast. Eratosthenes's results undertaken in 1ST century by Posidonius, were corroborated in Alexandria and Rhodes by the comparison between remarks is excellent.
Geometry, a cornerstone in modern civilization, also had its beginnings in Ancient Greece. Euclid, a mathematician, formed many geometric proofs and theories [Document 5]. He also came to one of the most significant discoveries of math, Pi. This number showed the ratio between the diameter and circumference of a circle.
Squaring the circle with a compass and straightedge had been a problem that puzzled geometers for years. In his notes under the drawing he recognized that “if you open your legs enough that your head is lowered by one-fourteenth of your height and raise your hands enough that your extended fingers touch the line of the top of your head, know that the centre of the extended limbs will be the navel, and the space between the legs will be an equilateral triangle” . This excerpt alone shows that Da Vinci had an immense understanding of proportion, as well as geometry. On this page, Da Vinci also wrote the exact proportions that he used, based his own observations and the ones used in Vitruvius’s book. Da Vinci
Pi, short for Piscine, meaning a rational source of water, is a rational man living in the irrational world, who believes in not one, but three religions, which some may say is irrational. Pi, whose family owned a zoo, faced many hardships
In addition, when Pi is in university and is introducing himself to each of his classes he uses repetition to explain his name. He says his name, writes it on the board, and underlines it. Pi uses ritual to get people in the habit of calling him Pi. This has significance to his past zoo life. Zoo animals need lots of care, this includes feedings, cleanings, and training. Pi is used to ritual, he knows that animals learn/live off of routine, and repetition, and so he has applied these skills to his classmates indicating a similarity between animals, and humans. Animals learn off of repetition, and routine, as do humans. Pi 's name has a mathematical link which has major symbolism to the entire novel. We all know that Pi is a large, and complicated number. Pi says in the novel, "That 's one thing I hate about my nickname, the way that number runs on forever." (Martel 316). I feel like the author included this quote to signify that Pi has been on a long journey, just like Pi says the numbers continue on. This quote was said towards the ending of the novel, and could represent the
» Part 1 Logarithms initially originated in an early form along with logarithm tables published by the Augustinian Monk Michael Stifel when he published ’Arithmetica integra’ in 1544. In the same publication, Stifel also became the first person to use the word ‘exponent’ and the first to indicate multiplication without the use of a symbol. In addition to mathematical findings, he also later anonymously published his prediction that at 8:00am on the 19th of October 1533, the world would end and it would be judgement day. However the Scottish astronomer, physicist, mathematician and astrologer John Napier is more famously known as the person who discovered them due to his work in 1614 called ‘Mirifici Logarithmorum Canonis Descriptio’.
Historically speaking, ancient inventors of Greek origin, mathematicians such as Archimedes of Syracuse, and Antiphon the Sophist, were the first to discover the basic elements that translated into what we now understand and have formed into the mathematical branch called calculus. Archimedes used infinite sequences of triangular areas to calculate the area of a parabolic segment, as an example of summation of an infinite series. He also used the Method of Exhaustion, invented by Antiphon, to approximate the area of a circle, as an example of early integration.
They constructed the 12-month calendar which they based on the cycles of the moon. Other than that, they also created a mathematical system based on the number 60 which they called the Sexagesimal. Though, our mathematics today is not based on their system it acts like a foundation for some mathematicians. They also used the basic mathematics- addition, subtraction, multiplication and division, in keeping track of their records- one of their contributions to this world, bookkeeping. It was also suggested that they even discovered the number of the pi for they knew how to solve the circumference of the circle (Atif, 2013).
Ever wonder how scientists figure out how long it takes for the radiation from a nuclear weapon to decay? This dilemma can be solved by calculus, which helps determine the rate of decay of the radioactive material. Calculus can aid people in many everyday situations, such as deciding how much fencing is needed to encompass a designated area. Finding how gravity affects certain objects is how calculus aids people who study Physics. Mechanics find calculus useful to determine rates of flow of fluids in a car. Numerous developments in mathematics by Ancient Greeks to Europeans led to the discovery of integral calculus, which is still expanding. The first mathematicians came from Egypt, where they discovered the rule for the volume of a pyramid and approximation of the area of a circle. Later, Greeks made tremendous discoveries. Archimedes extended the method of inscribed and circumscribed figures by means of heuristic, which are rules that are specific to a given problem and can therefore help guide the search. These arguments involved parallel slices of figures and the laws of the lever, the idea of a surface as made up of lines. Finding areas and volumes of figures by using conic section (a circle, point, hyperbola, etc.) and weighing infinitely thin slices of figures, an idea used in integral calculus today was also a discovery of Archimedes. One of Archimedes's major crucial discoveries for integral calculus was a limit that allows the "slices" of a figure to be infinitely thin. Another Greek, Euclid, developed ideas supporting the theory of calculus, but the logic basis was not sustained since infinity and continuity weren't established yet (Boyer 47). His one mistake in finding a definite integral was that it is not found by the sums of an infinite number of points, lines, or surfaces but by the limit of an infinite sequence (Boyer 47). These early discoveries aided Newton and Leibniz in the development of calculus. In the 17th century, people from all over Europe made numerous mathematics discoveries in the integral calculus field. Johannes Kepler "anticipat(ed) results found… in the integral calculus" (Boyer 109) with his summations. For instance, in his Astronomia nova, he formed a summation similar to integral calculus dealing with sine and cosine. F. B. Cavalieri expanded on Johannes Kepler's work on measuring volumes. Also, he "investigate[d] areas under the curve" ("Calculus (mathematics)") with what he called "indivisible magnitudes.
The history of math has become an important study, from ancient to modern times it has been fundamental to advances in science, engineering, and philosophy. Mathematics started with counting. In Babylonia mathematics developed from 2000B.C. A place value notation system had evolved over a lengthy time with a number base of 60. Number problems were studied from at least 1700B.C. Systems of linear equations were studied in the context of solving number problems.
The Golden Rectangle is a unique and important shape in mathematics. The Golden Rectangle appears in nature, music, and is often used in art and architecture. Some thing special about the golden rectangle is that the length to the width equals approximately 1.618……
A rectangle is a very common shape. There are rectangles everywhere, and some of the dimensions of these rectangles are more impressive to look at then others. The reason for this, is that the rectangles that are pleasing to look at, are in the golden ratio. The Golden Ratio is one of the most mysterious and magnificent numbers/ratios in all of math. The Golden Ratio appears almost everywhere you look, yet not everyone has ever heard about it. The Golden Ratio is a special number that is equal to 1.618. An American mathematician named Mark Barr, presented the ratio using the Greek symbol “Φ”. It has been discovered in many places, such as art, architectures, humans, and plants. The Golden Ratio, also known as Phi, was used by ancient mathematicians in Egypt, about 3 thousand years ago. It is extraordinary that one simple ratio has affected and designed most of the world. In math, the golden ratio is when two quantities ratio is same as the ratio of their sum to the larger of the two quantities. The Golden Ratio is also know as the Golden Rectangle. In a Golden Rectangle, you can take out a square and then a smaller version of the same rectangle will remain. You can continue doing this, and a spiral will eventually appear. The Golden Rectangle is a very important and unique shape in math. Ancient artists, mathematicians, and architects thought that this ratio was the most pleasing ratio to look at. In the designing of buildings, sculptures or paintings, artists would make sure they used this ratio. There are so many components and interesting things about the Golden Ratio, and in the following essay it will cover the occurrences of the ratio in the world, the relationships, applications, and the construction of the ratio. (add ...