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,
#(4-a)^2+(7-b)^2=r^2#.......#(1)#
#(3-a)^2+(4-b)^2=r^2#..........#(2)#
#(6-a)^2+(2-b)^2=r^2#.........#(3)#
We have #3# equations with #3# unknowns
From #(1)# and #(2)#, we get
#16-8a+a^2+49-14b+b^2=9-6a+a^2+16-8b+b^2#
#2a+6b=40#
#a+3b=20#.............#(4)#
From #(2)# and #(3)#, we get
#9-6a+a^2+16-8b+b^2=36-12a+a^2+4-4b+b^2#
#6a-4b=15#..............#(5)#
From equations #(4)# and #(5)#, we get
#6(20-3b)-4b=15#
#120-18b-4b=15#
#22b=105#, #=>#, #b=105/22#
#a=20-3*105/22=125/22#, #=>#, #a=125/22#
The center of the circle is #=(125/22,105/22)#
#r^2=(3-125/22)^2+(4-105/22)^2=(59/22)^2+(17/22)^2#
#=3770/484#
#=1885/242#
The area of the circle is
#A=pi*r^2=pi*1885/242=24.5#