question about circles?

Question

would it be safe to define a circle as a regular polygon with an infinite amount of sides?

Answer 1

close enough:

a circle is a regular pentagon with n equal sides, in the limit as n approaches infinity.

Answer 2

No. It's not a bad concept, but it's too sloppy for a definition.
To talk about "infinite amount" is always to tread on dangerous ground, though it can be done with care.

A correct statement would be that a sequence of regular polygons, with the distance from the centre to each vertex remaining constant, approaches a circle as the number of sides increases without limit (that's what "approaches infinity" means). The fixed distance is the radius of the circle. The perimeter and the area of the polygon approach the circumference and the area of the circle.

As for defining a circle, why try to improve on
"A co-planar set of points which are equidistant from a fixed point"??

Answer 3

No. It would, however, be safe to say that a regular polygon of n sides approaches a circle as the number of sides approaches infinity. The apothem of the polygon would approach the radius of the circle and the perimeter of the polygon would approach the circumference of the circle. The center of the regular polygon would be the exact center of the circle.

Answer 4

I'm sure that vlee meant to say 'polygon' not 'pentagon'.