# What is the travelled distance of the red mark on the upper surface of the rotating cube?

Each side of a cube is 2 unit in length. This cube is kept on a table such a way that one surface (i.e., 4 vertices) of it completely touches the table. At this position, a red point is drawn on the center of the upper surface. Now the cube is being rotated along a straight line towards a certain direction. At the time of rotation, at least two vertices of the cube are in touch with the table. Rotation is stopped when the red mark reached its initial position. Total distance traveled by the red mark is $$(\sqrt{{b}}+1)\pi$$

What is the value of b?

I am facing trouble to find the distance travelled by the red mark. Is it a straight line from start to end or a curved line (almost a circle) which is travelled by the red mark at every rotation of the cube?

The cube is rolled keeping one edge on the table at each time. As it is rolled around each edge the red point traces a quarter circle with radius the distance to the edge of rotation. For the first edge the point is $$\sqrt 5$$ away from the edge because we have a $$1-2-\sqrt 5$$ triangle from the center of the edge to the point, so the point moves $$\frac \pi 2 \sqrt 5$$. Then the point is $$1$$ from the edge of rotation for the next one, so it traces $$\frac \pi 2$$. The third edge has the point starting from the center bottom, again $$1$$ from the center, and we get another $$\frac \pi 2$$. The final one has the point $$\sqrt 5$$ away again, so we get a total of $$\frac \pi 2(2+2\sqrt 5)=\pi(1+\sqrt 5)\\b=5$$