To calculate the area of the circle, we must calculate the radius #r# of the circle
Let the center of the circle be #O=(a,b)#
Then,
#(2-a)^2+(4-b)^2=r^2#.......#(1)#
#(6-a)^2+(5-b)^2=r^2#..........#(2)#
#(4-a)^2+(3-b)^2=r^2#.........#(3)#
We have #3# equations with #3# unknowns
From #(1)# and #(2)#, we get
#4-4a+a^2+16-8b+b^2=36-12a+a^2+25-10b+b^2#
#8a+2b=41#.............#(4)#
From #(2)# and #(3)#, we get
#36-12a+a^2+25-10b+b^2=16-8a+a^2+9-6b+b^2#
#4a+4b=36#
#a+b=9#..............#(5)#
From equations #(4)# and #(5)#, we get
#(41-2b)/8=9-b#
#41-2b=72-8b#
#6b=72-41#, #=>#, #b=31/6#
#8a=41-2*31/6=41-31/3=92/3#, #=>#, #a=23/6#
The center of the circle is #=(23/6,31/6)#
#r^2=(2-23/6)^2+(4-31/6)^2=(-11/6)^2+(-7/6)^2#
#=170/36#
#=85/18#
The area of the circle is
#A=pi*r^2=85/18*pi=14.84#