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
Yonas Yohannes
Jan 21, 2016

#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:enter image source here
The idea is to express d and h only in terms of: #a, b, c, S_p#

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