The standard Cartesian form for the equation of a circle is:
(x−h)2+(y−k)2=r2 [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=r2 [2]
(8−h)2+(2−k)2=r2 [3]
(3−h)2+(6−k)2=r2 [4]
Expand the squares:
36−12h+h2+16−8k+k2=r2 [2.1]
64−16h+h2+4−4k+k2=r2 [3.1]
9−6h+h2+36−12k+k2=r2 [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=−92
Substitute h=−92 into equation [5] and then solve for k:
7−6(−92)+4k=0
34+4k=0
k=−172
Substitute h=−92 and k=−172 into equation [2] and the solve for r2:
(6+92)2+(4+172)2=r2
(122+92)2+(82+172)2=r2
(212)2+(252)2=r2
r2=5332
The area of the circle is:
Area=πr2
Area=5332π
