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A circle of radius R touches externally ...

A circle of radius R touches externally a set of 12 circles each of radius `r` surrounding it. Each one of the smaller circles touches two circles of the set. Then `R/r=sqrtm+sqrtn-1,` where `m,n in N and m+n` is

Text Solution

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Sum of interior angles of 12 sides polygon is=`(n-2)pi`
`=10pi`
`(10pi)/12=2(pi/2-theta/2)`
`(5pi)/6=pi-theta`
`theta=pi/6=30^0`
`(2r)^2=R^2+R^2-2Rcostheta`
`4r^2=2R(1-cos30^0)`
`R^2/r^2=1/(1-sqrt3/2)`
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