Triangle Perimeter

Adding three numbers is not where the interest lies. The real content of this page is which triples of numbers are allowed, and the rule is the triangle inequality: each side must be shorter than the other two put together. Lengths of 2, 3 and 10 will not close into a triangle no matter how you arrange them, and the reason is not a rule someone imposed — it is that the straight run from one corner to another is the shortest path between those two points, so detouring through a third corner can never be shorter. If $a + b$ exactly equalled $c$, the triangle would flatten into a line segment with no area at all, which is the boundary case the solver rejects.

The usual working direction is backwards. A surveyor closing a traverse, or anyone with a perimeter and two known sides, gets the third by subtraction: a triangular lot with 78 m of frontage in total, with two sides of 30 m and 26 m, has a third side of 22 m. That answer is legal — 22 + 26 exceeds 30 — but note that arithmetic alone would not have told you so, and a perimeter of 78 with sides of 30 and 46 gives a third side of 2 m that cannot be built.

Half the perimeter has a name and a life of its own. The semiperimeter $s = P/2$ is the gateway to Heron's formula, which extracts the area from the three sides alone, and to the incircle, whose radius is simply $r = A/s$. Both are reasons to compute a perimeter you did not think you needed. The triangle inequality itself outgrew geometry entirely: it is one of the three defining axioms of a metric space, so every notion of distance in mathematics — between functions, between signals, between DNA sequences — is required to obey the same rule these three fence lines do.

Two things to watch. Perimeter scales linearly with size while area scales as the square, so a lot with twice the dimensions needs twice the fencing but encloses four times the ground — fencing is cheap per hectare on large parcels and expensive on small ones. And the perimeter alone never determines the shape: an equilateral triangle of perimeter 78 m encloses 293 m², while a 30–26–22 triangle of the same perimeter encloses 285 m², and a nearly flat 38–38–2 encloses only 38 m². Same fence, very different paddock.