SOLUTION: A Regualr polygon of n sides inscribed in a circle of radius r has 3r squared. What is the value of n?

Question 32301: A Regualr polygon of n sides inscribed in a circle of radius r has 3r squared. What is the value of n?
Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
A Regualr polygon of n sides inscribed in a circle of radius r has 3r squared. What is the value of n?
YOU MEAN AREA =3R^2...ASSUMING SO...
THERE WILL BE N TRIAGLES WHEN N VERTICES ARE JOINED TO THE CENTRE OF THE CIRCUM CIRCLE.
SO EACH RIANGLE HAS AREA OF 3R^2/N
NOW AREA OF EACH TRIANGLE IS GIVEN BY 0.5*R^2*SIN(X)...WHERE X IS THE ANGLE SUBTENDED BY ONE SIDE OF REGULAR POLYGON AT CENTRE OF CIRCUMCIRCLE.
HENCE 0.5*R^2*SIN(X)=3R^2/N
SIN(X)=3/(N*0.5)= 6/N
NOW IF THERE ARE N SIDES THEN EACH SIDE SUBTENDS 360/N ANGLE AT CENTRE .HENCE
X=2PI/N...OR....N=2PI/X
SIN(X)=6*X/2PI=3X/PI
WE FIND THAT AT X=30...OR...PI/6,WE HAVE SIN(X)=SIN(30)=SIN(PI/6)=0.5
AND 3X/PI=(3*PI)/(6*PI)=0.5
HENCE X=PI/6 IS THE SOLUTION
HENCE N=2PI/X=(2PI)/(PI/6)=2*6=12
HENCE THE REGULR POLYGON IS 12 SIDED
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