Pi Case
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Pi essay
Pi is a mathematical constant which represents the ratio of a circles circumference compared to its diameter. Pi has been used throughout history in mathematical senses. However the symbol first arose as the 16th numeral of the Greek Alphabet. Because Pi is an irrational number, a number that cannot be expressed as a ratio containing two integers, it has often been estimated and rounded throughout time. One rounding technique is to utilize an inscribed and circumscribed polygon around a circle. When set up correctly all points of the inscribed polygon correlate to those of the circumscribed one. Using trigonometry the estimator can figure out the perimeter of each figure. When these perimeters are calculated they can be added together and divided by two. This will give you your estimation for Pi. These next three examples are basics of this estimation theory. The first will consist of a three sided polygon with 60 degree angles (triangle), the nest is a four sided polygon with four right angles (square), and lastly a hexagon consisting of five 72 degree angles. All three of these examples will show the accuracy of estimating Pi and how the accuracy changes with side length. We will be comparing our estimates to 3.141592654, or an abbreviated rounded form of the irrational number.
In this diagram the circumscribed triangle and inscribed triangle can be utilized to estimate Pi. The first step is to use trigonometry to find the perimeter of each triangle. You can do this by picking a side and setting up a right angle adjacent to that side. This gorgous newly formed triangle can be parted into an equation using TAN45. Once the equation is set (tan45= x/?) It can be multiplied by the number of sides. In this case our side length is multiplied by three. The same procedure is done for the circumscribed triangle. Once both these perimeters are figured out (which is P=6.75 and P=3.83) than they are added together to get 10.58. This numbered is divided by the number of sides (3) and the final estimation for Pi is 3.52. This is considerably close to our actual number for Pi.
Next we go through the same procedure with a square. Two squares are set in and out of a circle. Trigonometry is again used to figure out the side lengths still using TAN45 and the same equations. In this instance the perimeters come to P=8 and P=5.72. these together sums up to 13.72. This number must be divided by 4 to get our new estimation for Pi. This time it comes a little more accurate