Finding the Value of Π

I will presnt a way to prove π = 3.14159 ............ using geometry and trigonometry................

it was originally made by David Coulson, Teacher / writer in maths and science

The circumference of the circle is bigger than the perimeter of the inner hexagon,and is smaller than the perimeter of the outer hexagon...........................

The perimeter of the inner hexagon is exactly 6 radii..................

The perimeter of the outer hexagon is harder to work out.................

The length of each side is 2R Tan 30 2R Tan 30

So, the perimeter of the outer hexagon is

6 x 2R Tan 30 = 12R Tan 30

The circle is sandwiched between the inner hexagon and the outer hexagon.

6R < Circumference < 12RTan 30

c= 2 π R

therefore,

3 < π < 3.4

we can improve on the estimate by doubling the number of sides.

12 sin15 < π < 12 tan15

therefore,

3.106 < π < 3.21

In general, for an n-sided polygon inside the circle and another one outside of the circle,

n x Sin (180/n) < p < n x Tan (180/n)

As the number of sides increase,

the value of p is squeezed more tightly by the upper and lower limits.

SEE the chart above based on previous formula.........

the max and min valve of π is same when no.of sides = 6,114

and thus the value of π is 3.141593

BUT

When a calculator or spreadsheet generates the SIN or TAN of an angle, it is using a formula that depends on the value of pi.

You can use geometry to get the COS of 60 degrees.........that is 1/2 or 0.5 ....and then we can use the half angle formulas to generate all the SINs and TANs you need after that.

So starting from 60 degrees, we can calculate the Sin, Cos and Tan of increasingly small angles..............

from the image you can see the value of π to be

3.14159