Thread: 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?

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

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.

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.

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).

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.

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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 .
In either of the two smaller triangles .
That gives us .
Of course here .
Now you have the length of a side.

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