Are triangles with curved sides still considered triangles, in Euclidean space? So the sum of their angles can be bigger (or smaller) than 180 degrees? I can imagine triangles consisting of three curved, connected line pieces. Are they considered in mathematics?

As is suggested in a comment below, a [Reuleaux](https://en.wikipedia.org/wiki/Reuleaux_triangle) triangle is a triangle with curved sides. It has very specific sides though. Why not introduce general triangles of which the "normal" triangle and the Reuleax one are specific cases?

Don't curved triangles (of which the circle would be a special case, like all cone intersections) have any value? Like the distorted images of circles in cone intersections?