Q. ABC is a equilateral triangle  in which AO=BO=CO=r. Find the length of r.


Here OA = OB = OC = r so we need to calculate circum radius , also we know that joining O with B bisect angle B in equilateral triangle. {angle bisector}

and each angle is 60 degree so angle OBM = 30 degree
we have extended AO till M and we know that in equilateral triangle all circumcenter, orthocenter, centroid lies on same point so AM will be bisector of side BC and hence MB = a/2 
now in right angle triangle BMO 
cos30 = BMBO= a2rr = a2cos30= a2×32= a3

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