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it was originally made by David Coulson, Teacher / writer in maths and science

## 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

therefore,

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

therefore,

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

### 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.

## 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

3.14159

Runner Up in the

Pi Day Challenge 2016

## 8 Discussions

2 years ago

You are kidding, right? Pi is an irrational number that can not be represented exactly. It has an infinite number of decimal places. The value 3.1415926535 is only an approximation. The only reason you got 3.14159 is because of the precision of your arithmetic.

Reply 2 years ago

I will take an infinite sided polygon to make a circle....which makes the value of pi irrational.....thats how simple it is to comprehened.......nice work bro

Reply 2 years ago

He's using "..." to denote that it goes on. The proof is just for the first digits. And I think it's obvious that you need to go on to infinity to get a more accurate number.

Reply 2 years ago

Thanks for that reply.........I couldn't have given such a good reply.....your instructables are awesome........

Reply 2 years ago

ThomasK19 is right I meant that only I have not given the dots....... elsewhere but in the most significant place that is the title............but peabody1929 thanks for more precision..............

2 years ago

Wow i got my first prize ......Thanks instructables........

2 years ago

I like the "May I Have a Large Container of Coffee" trick :D Why didnt my math teacher tell me this

Reply 2 years ago

I guess internet is the best school and instructable is the best teacher...........