Question 1164064
<pre>
Let s=arc length, r=radius, and {{{alpha}}} = central angle (in radians)

We set up the problem with these two constraints:
The area of the given sector is (9*12)/2 = 54 {{{in^2}}}
The perimeter is 33in


So, 
    2r + s  =  33    (1)
     rs/2   =  54    (2)   
and arc-length relates to r and {{{alpha}}} by:
       {{{alpha}}}*r = s     (3)


Equations (1) and (2) can be reduced (by substitution) to one equation in variable r:
    2r + (108/r) = 33
    2r - 33 + 108/r = 0

Muliply by r:
    2r^2 - 33r + 108 = 0

The LHS factors to:
    (2r-9)(r-12) = 0    

Solutions:  r=9/2   and r=12  (discard r=12, as that is the already-known circle)

Plug in r=(9/2)in (=4.5in) into (1) to get s=24in, and then use these values for r and s in (3) to get {{{alpha = 48/9}}} rad  (=5.3 rad, to one tenth)


NOTICE that the radius of the first circle multiplied by 2 is the arc length found in the 2nd circle, and the arc length from the first circle divided by 2 is the radius of the 2nd circle.