A triangle has corners at #(3 ,1 )#, #(4 ,9 )#, and #(7 ,4 )#. What is the area of the triangle's circumscribed circle?

The three corners are #A (3,1) B (4,9) and C (7,4)#

Distance between two points #(x_1,y_1) and (x_2,y_2)# is

#D= sqrt ((x_1-x_2)^2+(y_1-y_2)^2#

Side #AB= sqrt ((3-4)^2+(1-9)^2) ~~ 8.06#unit

Side #BC= sqrt ((4-7)^2+(9-4)^2) ~~5.83#unit

Side #CA= sqrt ((7-3)^2+(4-1)^2) = 5.0# unit

The semi perimeter of triangle is #s=(AB+BC+CA)/2# or

#s= (8.06+5.83+5.0)/2~~ 9.45# unit

Area of Triangle is #A_t = |1/2(x1(y2−y3)+x2(y3−y1)+x3(y1−y2))|#

#A_t = |1/2(3(9−4)+4(4−1)+7(1−9))|# or

#A_t = |1/2(15+12-56)| = | -29/2| =14.5# sq.unit

Radius of circumscribed circle is #R=(AB*BC*CA)/(4*A_t)# or

#R=(sqrt(65)*sqrt(34)*sqrt(25))/(4*14.5) ~~ 4.05#

Area of circumscribed circle is #A_c=pi*R^2=pi*4.05^2~~51.6#

sq.unit [Ans]

The other answer gives an approximation for this question that may be answered exactly. I don't blame the other answer; we've all been taught to do this. I prefer the exact answer, which generally means avoiding square roots.

The circumcircle is just the circle through the three vertices; the triangle almost doesn't matter. Except, miraculously, the circumradius #r# equals the product of the triangle sides #a,b,c# divided by four times the triangle's area #A#.

#r = {abc}/{4A}#

It's much more useful squared, and we're looking for #pi r^2# anyway.

#pi r^2 = {pi a^2 b^2 c^2}/{16A^2}#

The coordinates give the squared distances easily. Archimedes' Theorem relates the squared distances to the triangle area:

# 16A^2 = 4a^2b^2 - (c^2-a^2-b^2)^2#

So,

#pi r^2 = {pi a^2 b^2 c^2}/{ 4a^2b^2 - (c^2-a^2-b^2)^2}#

We form the squared distances from pairs of points #(3,1),(4,9),(7,4)#

#c^2=(4-3)^2+(9-1)^2=65#

#a^2=(7-4)^2+(4-9)^2=34#

#b^2=(7-3)^2+(4-1)^2=25#

We're free to play with the assignment to #a,b# and #c.# I prefer making #c# the biggest one which gives me the smallest thing to square. (I try to do these by hand sometimes to keep in shape.)

#pi r^2 = {pi (65)(34)(25) }/{ 4(25)(34) - (65-34-25)^2} = (27625 pi)/1682 #

The other answer said #51.6,# this is #51.597203956847822956320... quad sqrt#