The standard Cartesian form for the equation of a circle is:
(x-h)^2+(y-k)^2=r^2" [1]"
where (x,y) is any point on the circle, (h,k) is the center point, and r is the radius.
The points (6,4), (8,2), and (3,6) must lie on the circumscribed circle, therefore, we can use these points to write 3 unique equations:
(6-h)^2+(4-k)^2=r^2" [2]"
(8-h)^2+(2-k)^2=r^2" [3]"
(3-h)^2+(6-k)^2=r^2" [4]"
Expand the squares:
36-12h+h^2+16-8k+k^2=r^2" [2.1]"
64-16h+h^2+4-4k+k^2=r^2" [3.1]"
9-6h+h^2+36-12k+k^2=r^2" [4.1]"
Subtract equation [4.1] from equation [2.1]:
7-6h+4k=0" [5]"
Subtract equation [4.1] from equation [3.1]:
23-10h+8k=0" [6]"
Multiply equation [5] by -2 and add it to equation [6]:
9+2h=0
h = -9/2
Substitute h = -9/2 into equation [5] and then solve for k:
7-6(-9/2)+4k=0
34 + 4k=0
k = -17/2
Substitute h = -9/2 and k = -17/2 into equation [2] and the solve for r^2:
(6+9/2)^2+(4+17/2)^2=r^2
(12/2+9/2)^2+(8/2+17/2)^2=r^2
(21/2)^2+(25/2)^2 = r^2
r^2 = 533/2
The area of the circle is:
"Area" = pir^2
"Area" = 533/2pi
