Although pi has been known for thousands of years, it wasn’t accurately calculated for many years and still mathematicians are calculating pi to more and more decimal places.

Archimedes was one of the first to calculate the value of pi more accurately. He did this by drawing a polygon inside a circle and one outside. Calculating areas of the 2 polygons found the upper and lower bounds for the area of the circle. For a circle of radius 1, the area is equal to pi and therefore pi can be estimated. For example:

The outside square has side length equal to the diameter of the circle. Therefore the area of the outside square will be 4cm^{2}. The inner square has a diagonal of 2cm. Using Pythagoras, the side length is calculated to be root 2. This makes the area 2cm^{2}.

Using a square, pi can be shown to be between the values 2 and 4. However, using polygons with more sides will improve the precision of the calculations.

## More sides:

Using an hexagon, the above method can be followed to estimate pi more accurately:

For the hexagon inside the circle, the shape can be split into 6 triangles. Each triangle is equilateral, all sides are 1cm and all angles 60 degrees. The area is found with 1/2*absinC. Each triangle has an area of about 0.433. Multiplied by 6, the area of the whole hexagon will be 2.598 (3dp).

The outer hexagon can be split into 12 triangles as shown in the above diagram (right). The angle is now 30 degrees. Trigonometry can be used to find the length of the opposite. Finding 1/2 x base x height gives the area of each individual triangle, which is 0.29. The area of the hexagon is therefore 3.48.

This shows pi is between 2.598 and 3.48, a smaller range than given by the square. A polygon with even more sides can be used to calculate pi more accurately than this.

## References:

http://uk.businessinsider.com/archimedes-pi-estimation-2014-3

https://www.exploratorium.edu/pi/history-of-pi