SOLUTION: Among all triangles with integer side length and perimeter 20 units, what is the area of the triangle with the largest area? Express your answer in simplest radical form.

Question 262992: Among all triangles with integer side length and perimeter 20 units, what is
the area of the triangle with the largest area? Express your answer in simplest radical form.

Answer by CharlesG2(834) (Show Source): You can put this solution on YOUR website!
Among all triangles with integer side length and perimeter 20 units, what is
the area of the triangle with the largest area? Express your answer in simplest radical form.
In geometry, Heron's (or Hero's) formula states that the area A of a triangle whose sides have lengths a, b, and c is

where s is the semiperimeter of the triangle:
. (this definition is taken from http://en.wikipedia.org/wiki/Heron's_formula the proof for it is there as well)
Not sure if you learned this formula or not, but am gonna use this to make the following table in Excel:
a b c s A
2 9 9 10 8.94427191
4 8 8 10 15.49193338
6 7 7 10 18.97366596
8 6 6 10 17.88854382
as you can see closer to an equilateral triangle the greater the area
The area is approximately 18.97 rounded to nearest integer 19
further explanation below:
If it was an equilateral triangle then a=b=c which from the above is 2*s,
and then 2*s = 3*a = 3*b = 3 * c and s = (3/2)a = (3/2)b = (3/2)c.
s-a = s-b = s-c or a=b=c
s-a = (3/2)a - a = (1/2)a

for example a 4 by 4 by 4 equilateral triangle would have area of 1/2 * base * height = 1/2 * 4 * sqrt(4^2 - 2^2) = 2 * sqrt(16 - 4) = 2 * sqrt(12) = 2 * 2 * sqrt(3) = 4 * sqrt(3), and A = 1/4 * 4^2 * sqrt(3) = 1/4 * 16 * sqrt(3) = 4 * sqrt(3)
in this case though the perimeter was 20, so if this were an equilateral triangle, each side would be 6 2/3 or 20/3
A = 1/4 * (20/3)^2 * sqrt(3) = 1/4 * 400/9 * sqrt(3) = 400/36 * sqrt(3)
A = 100/9 * sqrt(3) = approx. 19.245009 and that would be largest possible area

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