Heron's Formula

 triangle sss

Area of a Triangle from Sides

You can calculate the area of a triangle if you know the lengths of all three sides, using a formula that has been known for nearly 2000 years.

It is called "Heron's Formula" after Hero of Alexandria (see below)

 

Just use this two step process:

Step 1: Calculate "s" (half of the triangles perimeter):

s = a+b+c2

Step 2: Then calculate the Area:

herons formula A = sqrt( s(s-a)(s-b)(s-c) )

Example: What is the area of a triangle where every side is 5 long?

Step 1: s = 5+5+52 = 7.5
Step 2: A = √(7.5 × 2.5 × 2.5 × 2.5) = √(117.1875) = 10.825...Try it yourself:

Heron's Formula
Find a Triangle's Area from its Sides
Classic Heron's Formula:
s = (a+b+c)/2 = 6
Area = √( s(s-a)(s-b)(s-c) )
Area = 6
Variation with less rounding error:
Sides in Descending Order: 5,4,3
Area = √((a+(b+c))(c-(a-b))(c+(a-b))(a+(b-c)))/4
Area = 6
a:53.1301°
b:36.8699°
c:90°
Area is
6.000000
© 2018 MathsIsFun.com v0.61

 

xAS

Hero of Alexandria

The formula is credited to Hero (or Heron) of Alexandria, who was a Greek Engineer and Mathematician in 10 – 70 AD.

Amongst other things, he developed the Aeolipile, the first known steam engine, but it was treated as a toy!

Angles

In the calculator above I also used the Law of Cosines to calculate the angles (for a complete solution). The formula is:

triangle angle from sides C = cos^-1( (a^2+b^2-c^2)/2ab )

Where "C" is the angle opposite side "c".

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