An equilateral triangle rolls without slipping around a square of side length 1. See the picture. What's the length of the path that the marked point covers until the triangle and their point reach the starting positions for the next time? So here's the square and the triangle and the marked point. And so now let's just start rotating it around. We can think of this point as rotating about this point in a circle with radius 1. And so it rotates until it's here. So in total, this guy is 360 degrees. And now we remove this 90 degrees and this 60 degrees to calculate this angle right here, which is 210 degrees. So our point has just spun a total of 210 degrees to end up here. Well, now our point is, now our triangle is going to spin around this point. And so our point just isn't going to move and our triangle is going to go to here. And now finally, our triangle is going to start spinning around this point and our point is going to go back around to here and it's going to spin a total of 210 degrees again. So now let's think about what just happened. We spun 210 degrees twice and essentially moved 90 degrees backwards in a way, right? Because we started here and we ended up here, which is like a 90 degree rotation of our entire figure just in the opposite direction. So that means if we do this four times, we get back to our original position. So that means rotating a total of eight times 210 degrees is going to get us back to where we started. And so now we can calculate exactly what this is. Each one of these circles has radius 1. So it's circumference is 2 pi times the radius of 1. 210 degrees is what we're moving and we're doing this eight times for a total of 28 pi over 3.

equilateral triangle

rolling motion

square path

rotational movement

path length calculation