Question

formula between r and R in triangle​

Answer 1

Answer :-

Each central angle in a n-sided polygon = 360∘n=2πn

So, ∠AOB=2πn

Let the radius of circle circumscribing polygon  be R and radius of circle inscribed in the polygon be r.

From the figure,  OA= OB = R and OM = r

OM⊥AB

Consider the right triangle AMO,

∠AOM=12∠AOB=πn

AM = MB = a2

sin(πn)=a2R

R = a2sin(πn)   --------(1)

Similarly,  

tan(πn)=a2r

r = a2tan(πn)    ----------- (2)

Dividing eq(1) and eq(2):

Rr=a2sin(πn)a2tan(πn)

Rr=1cos(πn)

Note:While there also exists a simple relationship between R and r:

From the right triangle AMO,

R2=r2−(a2)2

or 4R2=4r2−a2

 

I hope it is helpful