Although the area formula for a triangle enables one to easily calculate the area of a triangular region, one must be able to measure the height (Equation 1).
In some cases, as in a farmerís field for instance, a measure of the height is difficult to obtain, although the individual side lengths are easily measured (Figure 1).
Figure 1: with side lengths, a, b and c
A calculation of area, given the individual side lengths is readily made with use of Heronís formula (Equation 2), where s, is the semi-perimeter, and is equal to one-half of the Perimeter (Figure 1).
Equation 3 represents one of the three relations from the Law of Cosines.
Solving Equation 1 for cos( A), we obtain Equation 4.
Equation 5 represents a fundamental trigonometric identity:
Solving Equation 3 for , we obtain Equation 6.
Substituting from Equation 4, we have Equation 7.
Squaring the term in parenthesis, we obtain Equation 8.
Rewriting 1 with a common denominator gives Equation 9.
Distributing the minus sign yields Equation 10.
Combining like-terms and factoring yields Equation 11.
Taking the square root of both sides gives Equation 12.
The area formula for a triangle is given in Equation 13.
In figure 2, we have , with height h and base, b, partitioned into segments and.
Figure 2: with height, h and partitioned base, b
From Figure 2 and trigonometric ratio of sides, we obtain Equation 14.
Substitution from Equation 14 into Equation 13 yields Equation 15.
Substituting from Equation 12, we obtain Equation 16.
Simplifying yields Equation 17.
Distributing and rearranging terms yields Equation 18.
We then write the equivalent relation given in Equation 19.
Rearranging terms yields Equation 20.
We rewrite one of the terms in the discriminate to obtain Equation 21.
We add zero to the discriminate via canceling terms, obtaining Equation 22.
We then factor the discriminate in Equation 22, yielding the simpler Equation 23.
We add zero again, via canceling terms, to the two factors of the discriminate obtaining Equation 24.
These additions of zero allow for the factoring of the discriminate in Equation 24, yielding the more elegant Equation 25.
Distributing the fraction under the radical, we obtain Equation 26.
We factor out the one-half from each factor of the radical obtaining Equation 27.
We define semi-perimeter (s) to be half of the perimeter in Equation 28.
Substitution from Equation 28 allows for the familiar expression of Heronís formula in Equation 29.
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