How many turns does a circle of circumference 1m make as it rolls around an equilateral triangle of perimeter 3m?

Question

A Math Olympiad question:

A circle of circumference $1$m rolls around an equilateral triangle of side $1$m (perimeter $3$m). How many turns does the circle make as it rolls around the triangle once without slipping?

Can anyone please help me with the question above?

The correct answer is 4 but I have no idea why it is 4 instead of 3. I have searched and found some information on the web about “the perimeter of the triangle rolls”. I have no idea what that means. Anyone can elaborate further?

Approximate image:

Thank you!

Answer 1

To count the turns, mark a fixed radius on the circle (in red below), then count how many times it turns around the center during the movement. While rolling along a side, the fixed radius turns at a constant (angular) rate, as expected. But watch what happens when you turn the corner. Right before the turn, the radius is perpendicular to the bottom side of the triangle. Right after the turn, the radius becomes perpendicular to the right side of the triangle. Therefore, the turn introduced an instantaneous additional rotation, and it's simple geometry to see that it's $120^\circ$. There is $3$ corners, so it adds up to one additional complete revolution by the time the circle rolls back to the starting position. Add that to the $3$ turns due to rolling, and it's the correct answer of $4$ turns total.

Once you visualize this right, it becomes obvious that there is nothing special about the triangle. If the circle rolled around a rectangle, instead, there would be $4$ turns of $90^\circ$ each, which would once again add up to one complete additional revolution. It doesn't even have to be a polygon, the same would happen around another circle, or around any simple closed curve, for that matter.

This additional revolution is the same reason why the moon actually rotates, even though we always see only one side of it from the earth (nicely illustrated in this animation on the NASA site). If your circle maintained the same point of contact with the triangle all along, so that there would be no rolling at all, it would still complete a full revolution just by sliding around the triangle.

Answer 2

At each corner the circle makes a third of a turn. From the position where it touches the corner and is tangent to the side of the triangle to the position "on top of the triangle" ( i.e. where the tangent is perpendicular to the height of the triangle) it makes a sixth of a turn, since from the picture You can see that it turns by an angle of $\alpha=60^{\circ}$. This happens again on the other side of the corner. So if the circle ends up in the same position as where it started it makes $3$ turns,$1$ per side, plus $3\cdot\frac{1}{3}$ for the corners, which makes $4$. However as @TonyK pointed out it might end at the other side of the corner as where it started and in this case it would have made just $3+\frac{2}{3}$ turns. The picture shows what happens when the circle passes from the position tangent to the side of the triangle to the top: