Friday, November 30, 2012
The use of Pi in geometry is very common. It is seen in formulas for area and circumference of circles, volume of cones and cylinders and more. Most of us know the value of Pi to be around 3.14, with many people memorizing Pi up to many digits. But where does the value of Pi come from?
The distance around the outside a circle is its circumference. Consider the formula for the circumference of a circle, Circumference = Pi x Diameter. Solving for Pi we get, Pi = Circumference divided by Diameter. Therefore, for a circle of any size, the circumference divided by the diameter is approximately 3.14. But how do we know this?
We can start approximating Pi but considering the perimeter of an regular n-sided figure and dividing that by the length of a diagonal. Take a square with side equal to one. By using the Pythagorean Theorem, or the knowledge of 45-45-90 triangles, the diagonal is approximately 1.41. Therefore the perimeter divided by the diagonal is approximately 2.82.
Now take an octagon, with sides of length equal to one. The length of the diagonal found by taking 2 times the apothem is approximately 2.61. Taking the perimeter of 8 divided by 2.61, we get an approximation for Pi to be 3.06. Notice that this is fairly close to the value of Pi or 3.14.
As the number of sides of a regular polygon increases, the shape more closely approximates that of a circle. When considering a regular polygon with 500 sides, each side of length one, we see calculate the diagonal to be approximately 159.155 and the approximation for Pi to be approximately 3.14157. This is extremely close to the value of Pi at 5 digits, which is 3.14159.
One can continue this process by increasing the number of sides of a regular polygon to get as close of an approximation as possible to the actual value of Pi. I have a book which has the value of Pi calculated out to 1 million digits. Why someone needs to know that, I don't know. For all practical purposes, 3.14 works fine. Also, the fraction 22/7 is often used in calculations involving Pi.