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Heron's Formula

heron.html

Exploration

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.

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 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

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

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

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 //.

{a+b+c->2*s,

a+c->2*s-b,

a+b->2*s-c,

-b-c->-(2*s-a)} //Simplify

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