## Proof of π and circumference of circle

## What is pi ?

pi is the the sixteenth letter of the Greek alphabet ( **Π**, **π** ), transliterated as ‘pi’. It’s approximate value is 3.14159265359 or 22/7.

In this blog , we try to find out the proof for the approximation of the value of pi (**Π**).

To approximate the value of pi , we will use the same kinda approach as in the previous post for the proof of area of circle.

## Proof of Π(pi)

Consider the below diagrams:

We know that the measure of angle of a polygon is 180(n-2)/n, where n is the number of sides of polygon. Also, when we look at figure II above,

we know that for a polygon (hexagon in above case), * ∠OBC ≅ ∠OCB* and it’s half of the interior angle of polygon because all internal triangles within a polygon are congruent and the radius bisects each of the interior angles of a polygon, thus we have

**m∠OBC = m∠OCB = 1/2 *(measure of interior angle of polygon) =**

Similarly we have * ∠BOX ≅ ∠XOC =1/2*m∠BOC –*————

**(a)**

Now * m∠BOC = 180-2*m∠OBC = 180-2*90*(n-2)/n = 360/n*—–

**(b)**

From (a) and (b) we have** m∠BOX = m∠XOC = 1/2*360/n = 180/n**

From the above** figures I and III**, we can clearly see that the perimeter of circle lies between the 2 polygons, as it’s sandwiched between them. Similarly, side of inner triangle is **2RSin30°**

Thus, collating all results we have for a hexagon we have

**6*2RSin30° < Circumference of circle < 6*2RTan30°**

further solving this, we have:-

** 6*2RSin30° < 2ΠR < 6*2RTan30° **

**=>6Sin30° < Π <6 Tan30°=> 3 < Π < 3.46. **

By increasing the number of sides of polygon this estimate can be improved and as a generic formula we have

for a **n-gon** * n*Sin(180/n) < Π < n*Tan(180/n)* —————-

**(c)**

In equation (c) above, say, for eg, we consider polygon with 180 sides and substituting n=180 , we get

**180*Sin(180/180) < Π < 180*Tan(180/180)**

=> **180*Sin1° < Π < 180*Tan1°**

=> *3.14143315871 < Π < 3.14191168708. *

As, we go on increasing the value of n, the closer we get to the approximation of pi. See the pi approximation table below:-

## Proof of circumference/perimeter of circle

Examining the proof of pi carefully, we can say that the circumference of circle lies between the circumference of the 2 polygons as seen in the above diagrams i.e * 2R*n*Sin(180/n) *and

*where*

**2R*Tan(180/n)***=*

**n***and*

**number of sides of polygon***.*

**R is the radius of circle**Also pi lies between* n*Sin(180/n) *and

*as proved above and hence we have circumference of circle =*

**n*Tan(180/n)****2πR.**