The History of Calculating Pi

From Archimedes's polygon divisons to Newton's calculus,one constant has interested mathematicians for about 2000 years! 

And it's non other than Pi 

  

 

So what's all the fuss about it?

Well, mathematicians  have been trying to calculate as many digits of it as accurately possible. Starting with Archimedes, he claculated that Pi would be between 3.1408 and 3.1429. Now that is all the accuracy you would need for Pi.However, mathematicians didn't stop there. For the next 2000 years, each contributed to higher and higher precisions. One astronomer, Christoph Grienberger, manually calculated 39 digits of Pi. How did they do this?

Well, it started with Archimedes division of polygons.

1.First you have to prove that Pi is between 3 and 4.

• To do this draw a hexagon inside of a circle with radius 1.

Circle with radius 1

• Then, divide the hexagon into 6 equilateral triangles.This way, you can prove that each side of the hexagon is(1) and that the perimeter of the hexagon is (6).

•  π=C/D ,or the ratio between the circumference and the diameter.Pi is also greater than the ratio between the perimeter of the hexagon and diameter of the circle, since the perimeter of the hexagon (6) is less than the circumference of the circle.
• Thus, π > 6/2 or 3.

2.To prove the pi is greater than 4, draw a square outside a circle with radius 1.

Length of each side is 2 and the perimeter of the square is 8

• Here pi is less than the ratio between the  perimeter of the square and the diameter of the circle since the perimeter of the square is larger than the circumference of the circle.
• Thus π < 8/2 or 4

We have finally managed to prove that 3<π<4.The next step is to narrow the range down.We do this by increasing the lowerbound and decreasing the upperbound.

This can be done by replacing the hexagon inside the circle with a larger polygon, a dodecagon for example, and then replacing the square outside the circle with a dodecagon.Now it will look something like this:

 In the end Archimedes reaches the 96-sided polygon inscribed and circumscribed on the circle, which narrowed down the range of the value of Pi to: 3.1409 < π < 3.1429.

From then on, mathematicians continued using Archimedes method on narrowing the range of Pi by inscribing and circumscribing higher and higher polygons on the circle.This continued for more than 2000 years until Newton invented calculus and created his own way of calculating the most elusive constant in Mathematics.