what is the name of a smooth curved line around the outside of a regular polygon
An apothem is a line drawn perpendicular to a side of a regular polygon from the center of the polygon. A polygon is not a circle so it cannot have a radius. The radius of a circle is drawn from the center to any point in the circumference of the circle. You can draw a circle which encloses the regular polygon touching all vertices. The polygon is said to be inscribed in the circle. The apothem will be less than the radius because the radius is not perpendicular to any side, it can be drawn to a vertex but the apothem is perpendicular to a side, so it is shorter. Ex: draw a square with a circle which inscribes it. You can see that the apothem will be less than the radius.
I assume you mean a polygon inscribed in a circle. It is regular if all its sides and angles are equal.
The circle has a smaller area than the polygon.
No because a regular polygon has equal sides and equal interior angles
what is the name of a smooth curved line around the outside of a regular polygon
A regular polygon
A regular polygon
An apothem is a line drawn perpendicular to a side of a regular polygon from the center of the polygon. A polygon is not a circle so it cannot have a radius. The radius of a circle is drawn from the center to any point in the circumference of the circle. You can draw a circle which encloses the regular polygon touching all vertices. The polygon is said to be inscribed in the circle. The apothem will be less than the radius because the radius is not perpendicular to any side, it can be drawn to a vertex but the apothem is perpendicular to a side, so it is shorter. Ex: draw a square with a circle which inscribes it. You can see that the apothem will be less than the radius.
I assume you mean a polygon inscribed in a circle. It is regular if all its sides and angles are equal.
The circle is an irregular shape because all of it is curved. * * * * * No. A circle is a regular shape. It is not a polygon and so cannot be a regular polygon but that does not prevent it from being a regular shape.
Inscribed Polygon
The circle has a smaller area than the polygon.
No because a regular polygon has equal sides and equal interior angles
A circle is non-polygon. Meaning it has a curved side. A circle has no sides. It is non-regular.
A regular polygon has a center, much as a circle does. There are also the sides which are all the same lengths. Then there is the apothem which is any segment that goes from the center and is perpendicular to one of the polygon's side. Then angles are also parts and they are all the same. You might consider the perimeter a part and of course it is the sum of the sides.
circumscribed means the polygon is drawn around a circle, and inscribed means the polygon is drawn inside the circle. See related links below for polygon circumscribed about a circle and polygon inscribed in a circle.