What is Heron's formula?

2 Answers
Gió
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

Dean R.
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)

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