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Heron's Formula

heron.html

Exploration

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

"heron_1.gif"

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

Approach

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.

Initialize

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Solution

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

Clear[a,b,c,s,cosA,sinA,E1];
cA=Solve[a^2==b^2+c^2-2*b*c*cosA,cosA] //Simplify

"heron_3.gif"

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

"heron_4.gif"

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

"heron_5.gif"

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}]

"heron_6.gif"

Simplify to Heron's formula.

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

"heron_7.gif"


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