How round is your circle? seems like a very strange question. To an engineer however, every physical object is only an approximation to a geometric ideal. Hopefully, this has been manufactured to a specified tolerance. Round parts are particularly important, and the task now is to assess departure from roundness.
The width of a smooth shape is the distance between parallel tangents. When a shape is not smooth we must talk about parallel support lines instead.
If we have a circle then the width is constant
If the width is constant, then is the shape a circle?
The construction for Reuleaux's rotor can be made more general. Any odd regular polygon can be used as a base. The sharp corners can be smoothed off by expanding the shape and adding other smaller circular arcs.
The above generalizations all use an odd polygon as a base. This family of shapes is based on an isosceles triangle. The right angles isosceles triangle has important applications later.
Indeed, symmetries are not needed. Any triangle can be the base of a shape of constant width. Having constant width is not a special trick for regular polygons.
Circular arcs are not necessary either, although they are much easier to draw.
In a square of width w draw a convex curve from top to bottom touching the left hand side, and which is tangent to the top and bottom of the square. At no point should the curvature be less than the curvature of a circle with radius w. Complete the curve eccentrically to create a shape of constant width, by taking a normal to the curve of length w. This can be done by hand with a ruler very quickly, and reasonably accurately.
Difficult to describe, but have a look at the shape generated by completing half an ellipse.