# Thread: Proof of regular polyon inscribed in a cricle

1. ## Proof of regular polyon inscribed in a cricle

Hello ,

I wanted to find a relationship between the number of sides of a regular polygon inscribed in a unit circle and the length of side of the polygon (The vertices are roots of equation in the form of Z^n=1).

I came up with two equation that Length = 2sin(pi/n) and Length= Sqrt(2-2cos(2pi/n) , where n is the number of sides.

I was able to prove the two roles geometrically ,

But I need to prove them , or at least one of them analytically (e.g. using algebra)

Can you help me?

2. ## Re: Proof of regular polyon inscribed in a cricle

Originally Posted by Hyunqul
Hello ,

I wanted to find a relationship between the number of sides of a regular polygon inscribed in a unit circle and the length of side of the polygon (The vertices are roots of equation in the form of Z^n=1).

I came up with two equation that Length = 2sin(pi/n) and Length= Sqrt(2-2cos(2pi/n) , where n is the number of sides.

I was able to prove the two roles geometrically ,

But I need to prove them , or at least one of them analytically (e.g. using algebra)

Can you help me?
I really don't have the patience to write it all out again. Read Lesson 7 in this thread...

Harry Potter Forums :: The Slug Club - Arithmancy Master Class (Private) :: HP Community & Forums

3. ## Re: Proof of regular polyon inscribed in a cricle

Pardon ?
May I repeat my questions ... I'm looking for porving the relationship bewteen the number of sides and length of one side of a regular polygon inscribed inside a unit circle of radius (1). The rules I came with were : Length = 2sin(pi/n) and Length= Sqrt(2-2cos(2pi/n). I've made the gemoetric proofs....and looking for alegbriac proofs....Do they exist in the link you offered me ? Can you instruct me to their location ?
thanks.

4. ## Re: Proof of regular polyon inscribed in a cricle

Originally Posted by Hyunqul
Pardon ?
May I repeat my questions ... I'm looking for porving the relationship bewteen the number of sides and length of one side of a regular polygon inscribed inside a unit circle of radius (1). The rules I came with were : Length = 2sin(pi/n) and Length= Sqrt(2-2cos(2pi/n). I've made the gemoetric proofs....and looking for alegbriac proofs....Do they exist in the link you offered me ? Can you instruct me to their location ?
thanks.
Look for Lesson 7, it's on that page... If you understand what is being done in that lesson, then a single extra step will complete your proof... It will not involve sines or cosines.

5. ## Re: Proof of regular polyon inscribed in a cricle

Originally Posted by Hyunqul
length of one side of a regular polygon inscribed inside a unit circle of radius (1). The rules I came with were : Length = 2sin(pi/n) and Length= Sqrt(2-2cos(2pi/n).
Have a look at this webpage.
Look at equation (6).

6. ## Re: Proof of regular polyon inscribed in a cricle

Hmmmm..I'm sorry Gentelmen , I cant quiet get what you presented.

If you could provide more calrification of ALGEBRAIC PROOF ... That would be vey generous of you ...otherwise, thanks anyway.

7. ## Re: Proof of regular polyon inscribed in a cricle

\
Originally Posted by Hyunqul
If you could provide more calrification of ALGEBRAIC PROOF ... That would be vey generous of you ...otherwise, thanks anyway.
In the diagram I have added a perpendicular bisector of the cord a.
That happens to also bisect the angle $\theta$.
In either of the two smaller triangles $\sin\left(\frac{\theta}{2}\right)=\frac{a}{2R}$.
That gives us $a=2R\sin\left(\frac{\theta}{2}\right)$.
Of course here $R=1~\&~\theta\text=\left(\frac{2\pi}{n}\right)$.
Now you have the length of a side.

8. ## Re: Proof of regular polyon inscribed in a cricle

hmmmm...thanks sir , but my aim was an algebraics proof....a mostly analytical one (e.g.using trig identities).