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Heron's Formula



Show that the area, K, of a ΔABC is given by


where the semi-perimeter s=(a+b+c)/2 and a, b and c are the lengths of the sides.


In a ΔABC with side lengths a, b and c, derive an expression for cos A (the cosine of the angle at vertex A of the triangle) using the Law of Cosines.  Using the identity "heron_2.gif" the area can be computed using K=(1/2)b c sin A.  Simplify the resulting expression for the area, K, to Heron's formula.


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Find an expression for the cos A using the Law of Cosines.

cA=Solve[a^2==b^2+c^2-2*b*c*cosA,cosA] //Simplify


Find an expression for the sin A using the previous expression for cos A.  Use the positive result.

sA=Solve[(sinA^2+cosA^2==1) /. cA,sinA] //Last


sA=Solve[(sinA^2+cosA^2==1) /. cA,sinA] //Last


Compute the area of the triangle from one-half the product of the base and height.

K1=FullSimplify[(b*c*sinA/2 /. sA) /. Sqrt[E1_]:>Sqrt[Together[E1]],Assumptions->{a>0,b>0,c>0}]


Simplify to Heron's formula.

K2=K1 //.
    -b-c->-(2*s-a)} //Simplify


Copyright © 1999-2007 Donald L. Vossler, Descarta2D Publishing