What is Heron's formula?

2 Answers
Jan 12, 2015

Heron's formula allows you to evaluate the area of a triangle knowing the length of its three sides.
The area #A# of a triangle with sides of lengths #a, b# and #c# is given by:

#A=sqrt(sp×(sp-a)×(sp-b)×(sp-c))#

Where #sp# is the semiperimeter:

#sp=(a+b+c)/2#

For example; consider the triangle:
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The area of this triangle is #A=(base×height)/2#
So: #A=(4×3)/2=6#
Using Heron's formula:
#sp=(3+4+5)/2=6#
And:
#A=sqrt(6×(6-5)×(6-4)×(6-3))=6#

The demonstration of Heron's formula can be found in textbooks of geometry or maths or in many websites. If you need it have a look at:
http://en.m.wikipedia.org/wiki/Heron%27s_formula

Jun 15, 2018

Heron's Formula is usually the worst choice for finding the area of a triangle.

Explanation:

Alternatives:

Area #S# of a triangle with sides #a,b,c#

#16S^2=(a+b+c)(-a+b+c)(a-b+c)(a+b-c)#

Area #S# of a triangle with squared sides #A,B,C#

#16S^2 = 4AB-(C-A-B)^2=(A+B+C)^2-2(A^2+B^2+C^2)#

Area of a triangle with vertices #(x_1, y_1), (x_2, y_2), (x_3, y_3)#

#S = 1/2 | (x_1- x_3)(y_2 - y_3) - (x_2 - x_3)(y_1 - y_3)| = 1/2 | x_1 y_2 - x_2 y_1 + x_2 y_3 - x_3 y_2 + x_3 y_1 - x_1 y_3 |#

Oh yeah, Heron's Formula is

#S = sqrt{s(s-a)(s-b)(s-c)}# where #s=1/2(a+b+c)#