How far can you swing open a circular segment?

Question

Cut a circle into a segment, then swing open that segment. How far can it go?

I constructed this in MS Paint and it looks like the answer is always 180 degrees. However, I cannot be sure the two pieces are not squishing into each other.

I would like a more concrete proof. I believe the max angle will be where the segment and original curve are tangent, but I'm not sure of that either.

This is how I'm thinking of it: The perfect hinge does not contact anything. So it could rotate 360 degrees as long as the "levers" don't bump into each other. A tangent line has only one point of contact. Therefore, to stop the rotating part, some other part of the piece has to contact some other part of the original circle.

Is the answer gonna depend on the length of the chord? If so then we have to get an answer in terms of chord/diameter ratio.

Answer 1

Nice observation and investigation.

Assuming you have a "perfect" hinge, swinging on a point and not deforming in any way, then considering the tangent line (at the closed hinge) leads to your answer. Rotating the top part of the tangent line along with the segment, you see it can rotate 180° and then the two parts of the tangent line touch, beyond which point you would get overlap of the segments. Since the circle is entirely to one side of the tangent, there is no overlap before that angle.

Suppose that the hinge is rotated by some small angle $\delta$ more than $180°$. Then the portions of the circle immediately adjacent to the hinge will overlap since the arc of the "lid" will start back from the hinge at an angle of $\delta$ inside the arc of the "body" that leads to the hinge. So $180°$ is the maximum rotation also.

Answer 2

$\alpha+ \beta $ always $180^0$ in your sketch.Angle between segment line was zero, became $180^0$ in full open condition when tangents contact.

EDIT1

The minor segment could jolly well rotate beyond 180 degrees geometrically and have intersection points below. It is a common hinged bottle cap design experience that we cannot open out beyond a point as the bottle body itself comes in the way. Neither is it needed. Else to verify the situation take a thin plastic disk and cut it into two segments, one major and one minor, leaving behind a small point to hinge it out. The material which you assumed existed (but not stated to be physical, but only geometrical) hinders further anticlock rotation. Unless the solid part had a ghost capability of permitting further interference rotation all out..