Question

How do you use Heron's formula to find the area of a triangle with sides of lengths `12`, `15`, and `18`?

Answer 1

`"Area"_triangle= 135/4sqrt(7)~~89.3` sq.units

Explanation:

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`