The area ratio = polygonArea / minBoundingCircleArea. For example the value could be 0.7, 0.9, etc. If the polygon is close to the circle, then the area ratio will be close to 1.
The perimeter ratio = polygonShape_length / minBoundingCircleShape_length. For example the value could be 1.5, 2.8, etc. If the polygon is close to the circle, then the perimeter ratio will be close to 1. It is not enough to evaluate area only; you need to take into account both ratios.
For the second part, can you describe what the constraints are?