Area of a Triangle as a Function of the Lengths of its 3 Sides

[Uncwilly]

Am currently thinking about triangles with sides of:
1, 1, 1.9999 (or 2)
1, 1, 1.4142
and
1, 1, 0.0001 (or 0)

These seem to the best cases to ponder.
My marginal time is not expansive enough to work through this at the moment.

[henryzz]

An interesting extension to the problem is to find a formula for the maximal area of a cyclic polygon given n lengths of sides.

[Dr Sardonicus]

Originally Posted by Dr Sardonicus

I learned this in high school geometry class.

Some things can be discerned without knowing the answer. Assume there is an algebraic formula F(a, b, c). Then

1) F(a, b, c) = F(a, c, b) = F(b,c,a) = F(b, a, c) = F(c, a, b) = F(c, b, a)

2) If k is a positive constant, then F(k*a, k*b, k*c) = k^2 * F(a, b, c)

3) Assuming 0 < a <= b <= c, and c = a + b, then F(a, b, c) = 0.

4) If 0 < a <= b <= c, and c < a + b, then F(a, b, c) is not a positive real number.

My legendary typing abilities strike again! I used the wrong inequality sign in the last statement. Of course, I should have said,

4) If 0 < a <= b <= c, and c > a + b, then F(a, b, c) is not a positive real number.

The condition may be re-expressed, "If 0 < a <= b <= c, and (a + b + c)/2 < c..."

[moeses]

I was reading about the solution to this problem and was curious why the solution needs to use the semi perimeter at all? It seems like once you have the area in terms of sides a, b, c then you’re done. Why introduce the semi perimeter? Is there an advantage?


https://en.m.wikipedia.org/wiki/Heron%27s_formula

[henryzz]

Originally Posted by moeses

I was reading about the solution to this problem and was curious why the solution needs to use the semi perimeter at all? It seems like once you have the area in terms of sides a, b, c then you’re done. Why introduce the semi perimeter? Is there an advantage?


https://en.m.wikipedia.org/wiki/Heron%27s_formula

It makes it much easier to remember. Try memorizing the first of the alternative formulations. Not so easy.

[Dr Sardonicus]

A number of formulas using the semiperimeter (particularly of a triangle) are listed on the Wikipedia page on the Semiperimeter. The "law of cotangents" might be of particular interest.

[rudy235]

I would never remember the formula of the area of a triangle in this particular case if I did not use the semiperimer.