A triangle has corners at $(1 ,3 )$ , $(2 ,5 )$ , and $(6 ,4 )$ . What is the area of the triangle's circumscribed circle?

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Answer 1 · 2016-09-05T07:14:01

Area of circumscribed circle is $21.4288$

If the sides of a triangle are $a$ , $b$ and $c$ , then the area of the triangle $Delta$ is given by the formula

$Delta=sqrt(s(s-a)(s-b)(s-c))$ , where $s=1/2(a+b+c)$

and radius of circumscribed circle is $(abc)/(4Delta)$

Hence let us find the sides of triangle formed by $(1,3)$ , $(2,5)$ and $(6,4)$ . This will be surely distance between pair of points, which is

$a=sqrt((2-1)^2+(5-3)^2)=sqrt(1+4)=sqrt5=2.2361$

$b=sqrt((6-2)^2+(4-5)^2)=sqrt(16+1)=sqrt17=4.1231$ and

$c=sqrt((6-1)^2+(4-3)^2)=sqrt(25+1)=sqrt26=5.0990$

Hence $s=1/2(2.2361+4.1231+5.0990)=1/2xx11.4582=5.7291$

and $Delta=sqrt(5.7291xx(5.7291-2.2361)xx(5.7291-4.1231)xx(5.7291-5.0990)$

= $sqrt(5.7291xx3.4930xx1.6060xx0.6301)=sqrt20.2507=4.5001$

And radius of circumscribed circle is

$(2.2361xx4.1231xx5.0990)/(4xx4.5001)=2.6117$

And area of circumscribed circle is $3.1416xx(2.6117)^2=21.4288$

A triangle has corners at (1 ,3 )(1 ,3 ), (2 ,5 )(2 ,5 ), and (6 ,4 )(6 ,4 ). What is the area of the triangle's circumscribed circle?