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?

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?

1 Answer

Impact of this question

2251 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-01-21T21:58:44

$A = \sqrt{S_{p} (S_{p} - a) (S_{p} - b) (S_{p} - c)}$ $A = sqrt(S_p(S_p-a)(S_p-b)(S_p-c))$

$A = \sqrt{19.2 (\frac{19}{2} - 4) (\frac{19}{2} - 7) (\frac{19}{2} - 8)}$ $A = sqrt(19.2(19/2-4)(19/2-7)(19/2-8) )$

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:
$A = \frac{1}{2} b h$ $A = 1/2 bh$ but why not use only the sides. Well if you decide to do that then welcome to Heron formula:

$A = \sqrt{S_{p} (S_{p} - a) (S_{p} - b) (S_{p} - c)}$ $A = sqrt(S_p(S_p-a)(S_p-b)(S_p-c))$

Where $S_{p}$ $S_p$ is the semi-perimeter:

$S_{p} = \frac{a + b + c}{2}$ $S_p= (a+b+c)/2$

There is an elegant prove to this leveraging trigonometry and using the following identities:
$sin θ = \sqrt{1 - cos θ} = \frac{\sqrt{4 a^{2} b^{2} - {(a^{2} + b^{2} - c^{2})}^{2}}}{2 a b}$ $sintheta = sqrt(1-costheta) = (sqrt(4a^2b^2-(a^2+b^2-c^2)^2))/(2ab)$
$A = \frac{1}{2} b h = (\frac{1}{2} a b) sin θ$ $A = 1/2bh = (1/2ab)sintheta$

Try to complete the derivation...

You can also leverage "Pythagorean Theorem" on the following triangle... see image: enter image source here
The idea is to express d and h only in terms of: $a, b, c, S_{p}$ $a, b, c, S_p$

Science

Math

Humanities

... and beyond

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?

1 Answer

Explanation:

Related questions

Impact of this question