Are triangles the strongest shape?

Solution 1:

As you asked about the strength of a triangular shape then let me introduce to the triangular chain consisting of three rigid links or bars connected to each other by pin joints(allowing rotation between two joined links) .

The degree of freedom (n) of a plane chain is given by the Grasshoff's law as $$n=3(l-1)-2j-h$$ for a triangular chain we have $$l=\text{no. of links}=3$$ $$j=\text{no. of binary joints}=3$$ $$h=\text{no. of higher pairs}=0$$ Hence, we get $$n=3(3-1)-2(3)-0=6-6=0$$ The degree of freedom of the triangular chain (equivalent to plane triangular shape) has zero degree of freedom this indicates that links of the triangular chain can't move even a bit if links are strong enough even under the application of external forces.

Thus a triangular shape is the strongest one which is also called a rigid structure. It is also called a perfect frame in physical structures.

Solution 2:

Here's one part of it.

As far as polygons go, a triangle is the only one that is defined by its side lengths. If you have a triangle of sides 5,6, and 7, there is only one shape it can take. The same cannot be said of other polygons. Imagine a square. It can be squished into a diamond with the same side lengths.

There is SSS congruence for triangles, but no analogous congruence for other polygons.

That's what diagonal bracing does in physical structures. Creates triangles.