How far can you swing open a circular segment?

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.

• Hint: draw the tangent at one of the points, then look at the point reflection of the figure across that point.
– dxiv
Feb 27 '17 at 3:37
• @DrZ214 btw how did you draw the center sketch in Paint? I mean how is the partition line rotated by about 150 degrees? Feb 27 '17 at 4:17
• @Narasimham Hehe, I just draw a straight line then a curved line using the curved line button. In other words, I just approximately sketched it. If I was less lazy, I wuda opened it in a more powerful editor that can do rotations with aliasing in any angle. But I hate opening those because they take forever, whereas paint opens right away. Feb 27 '17 at 4:19

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.

• Rotating it 180 degrees definitely means the two tangent lines coincide again. But to be rigorous, at 180 degs, how can we tell that some other part of the circle will not squish into the original circle? Feb 27 '17 at 3:48
• I tried to anticipate that question by pointing out that the entire circle is to one side of the tangent line. Feb 27 '17 at 3:51
• The way I think of it, the perfect hinge could rotate 360 degrees if it doesnt contact anything. "The hinge does not contact itself." It's the other parts of the circle that make contact and thus prevent further rotating. Feb 27 '17 at 4:17
• Yes, the "perfection" I was interested in was swinging on a point, no stretch etc. Feb 27 '17 at 4:18
• Sorry but I have to undo the accepted answer. I edited the OP with the way I'm thinking of it. You are right when you say no part of the circles go beyond the tangent line. However, if contact is only on the hinge, then no part of the "levers" bump into each other and prevent further rotation. If somehow the segment cannot rotate further, I need a proof or good reason why. Feb 27 '17 at 4:24

$\alpha+ \beta$ 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..

• Yes, a 180 degree would meant the tangent lines coincide again. But why does that prevent further rotation? If contact is only at the hinge, it can still swing further. It's contact some point away from the hinge, where the two "levers" bump into each other, that stops the rotation. Feb 27 '17 at 4:25
• Edited in the answer above. They call it living hinge used for tomato ketchups, cooking oil &c.. Feb 27 '17 at 8:03
• Your edit doesn't explain much. In my setup, the hinge is just a point on the circle. There is no extension. The pics in your link suggest that a living hinge has an extension. In fact, if you scroll down far enough, you'll see the diagram that clearly shows a living hinge does have an extension. My circle parts don't. I used the word "lever" to describe the circle segments as one swings out. Picture a door without handles. It won't stop swinging until a part downrange from the hinge bumps into the wall. The hinge itself contacting the wall (tangent point) should not stop anything. Feb 27 '17 at 12:05
• In a lever there are 3 points,Point of force application, hinge and reaction point. The last two merge in case of tangent.equilibrium. Feb 27 '17 at 12:55