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it was originally made by David Coulson, Teacher / writer in maths and science
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Step 1: Basic Diagram
The circumference of the circle is bigger than the perimeter of the inner hexagon,and is smaller than the perimeter of the outer hexagon...........................
Step 2: Inner Hexagon
The perimeter of the inner hexagon is exactly 6 radii..................
The perimeter of the outer hexagon is harder to work out.................
Step 3: Outer Hexagon
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
Step 4: Value of Pi
The circle is sandwiched between the inner hexagon and the outer hexagon.
6R < Circumference < 12RTan 30
c= 2 π R
3 < π < 3.4
Step 5: More Detailed
we can improve on the estimate by doubling the number of sides.
Step 6: More Detailed Vale of Π
12 sin15 < π < 12 tan15
3.106 < π < 3.21
Step 7: FORMULA
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.
Step 8: Final Value
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
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.
Step 9: Solving Problem
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.
Step 10: Value Found to Be 3.14159
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
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