Question

How do you use Heron's formula to determine the area of a triangle with sides of that are 4, 7, and 8 units in length?

Answer 1

`A = sqrt(S_p(S_p-a)(S_p-b)(S_p-c))`

`A = sqrt(19.2(19/2-4)(19/2-7)(19/2-8) )`

Plug in your calculator...

Explanation:

Heron formula requires you know only the sides of a triangle to compute the area. Note you can use another approach i.e. determine the height of the triangle and use our familiar:
`A = 1/2 bh` but why not use only the sides. Well if you decide to do that then welcome to Heron formula:

`A = sqrt(S_p(S_p-a)(S_p-b)(S_p-c))`

Where `S_p` is the semi-perimeter:

`S_p= (a+b+c)/2`

There is an elegant prove to this leveraging trigonometry and using the following identities:
`sintheta = sqrt(1-costheta) = (sqrt(4a^2b^2-(a^2+b^2-c^2)^2))/(2ab)`
`A = 1/2bh = (1/2ab)sintheta`

Try to complete the derivation...

You can also leverage "Pythagorean Theorem" on the following triangle... see image:
The idea is to express d and h only in terms of: `a, b, c, S_p`