I guess this answers my question before, though one thing to consider (this is for more standard polygon shapes) but you can build many interior shapes with triangles which makes them useful for that. (also related to proof that the sum of angles in a shape is related to the number of sides times 180 degrees)

KaiAmelung

Are there shapes that are not planar or without a well-defined interior? I suppose if you have a shape with greater than 3 sides you could have angles greater than 180 degrees which would make things messy, but the interior is still well-defined in this case, at least I think. I also guess I assumed "shapes" are 2d so they are all guaranteed to be planar but maybe that isn't technically the case.

yirenng

@KaiAmelung -- in 3D, imagine a quadrilateral defined by four vertexes that do not lie in the same plane. What is the set of points in the "interior" of this quadrilateral? Compare this to the triangle case, where in 3D the three vertexes of the triangle are guaranteed to lie in a plane.

Also, even in 2D, a polygon with a boundary composed of many vertexes may have a boundary that loops on itself -- in this case, one needs to add mathematical rules to define which points are "interior" to the polygon.

I guess this answers my question before, though one thing to consider (this is for more standard polygon shapes) but you can build many interior shapes with triangles which makes them useful for that. (also related to proof that the sum of angles in a shape is related to the number of sides times 180 degrees)

Are there shapes that are not planar or without a well-defined interior? I suppose if you have a shape with greater than 3 sides you could have angles greater than 180 degrees which would make things messy, but the interior is still well-defined in this case, at least I think. I also guess I assumed "shapes" are 2d so they are all guaranteed to be planar but maybe that isn't technically the case.

@KaiAmelung -- in 3D, imagine a quadrilateral defined by four vertexes that do not lie in the same plane. What is the set of points in the "interior" of this quadrilateral? Compare this to the triangle case, where in 3D the three vertexes of the triangle are guaranteed to lie in a plane.

Also, even in 2D, a polygon with a boundary composed of many vertexes may have a boundary that loops on itself -- in this case, one needs to add mathematical rules to define which points are "interior" to the polygon.