Heron's formula says that a triangle with sides of length, #a, b, c# has an area:
#color(white)("XXX")"Area"_triangle=sqrt(s(s-a)(s-b)(s-c))#
where #s# is the semi-perimeter #=(a+b+c)/2#
Given sides #12, 15, 18#
#color(white)("XXX")s=45/2#
and
#color(white)("XXX")"Area"_triangle=sqrt(45/2 * (45/2-12) * (45/2-15) * (45/2-18)#
#color(white)("XXXXXXX")=sqrt(45/2) * (21/2) * (15/2) * (9/2))#
#color(white)("XXXXXXX")=sqrt(((3 * 3 * 5) * (3 * 7) * (3 * 5) * * (3 * 3))/(2 * 2 * 2 * 2))#
#color(white)("XXXXXXX")=sqrt(((3^3)^2 * 5^2 * 7)/((2^2)^2))#
#color(white)("XXXXXXX")=(3^3 * 5)/2 sqrt(7)#
#color(white)("XXXXXXX")=135/4 sqrt(7)#
#color(white)("XXXXXXX")~~89.3#
