Question

An equilateral triangle with side `s` is "rolled" sideways twice. How far has each vertex traveled, given they travel in a circular arc?

Answer 1

Two vertices move `2/3 pi s` and one moves `4/3 pi s.`

Explanation:

I reworded this question that has been sitting around for a few months.

Let's call the triangle ABC with an apex up top of A and a base on the floor of BC, B to the left of C.

We'll roll right, keeping C stationary, rotating B so it becomes the new apex and A becomes the vertex on the floor along with C.

Clearly A and B moved `1/3` of the circumference of a circle of radius `s` so `2/3 pi s.`

Now we rotate around A. C moves to the apex, B to the floor, each traveling `2/3 pi s.`

So in total A and C moved `2/3 pi s` and B moved `4/3 pi s`