 The Rolling Triangle

• ## The Rolling Triangle

PoW ID: 3375

6th to 8th, High School, Geometry, Geometry, Geometry Library

### Problem Print

Equilateral triangle ABC is sitting on the top of square DEFG so that AC is coincident with DE. The triangle and square both have edges one unit in length.

Rotate the triangle around point C until point B is coincident with point F and the triangle is on the right side of the square. Call this one "roll".  Roll the triangle again, this time with B as the center of rotation, putting the triangle on the bottom of the square. Keep rolling until the triangle is back on top of the square.

What is the length of the path traced out by point A?

Extra: Do the same process, but keep going until the triangle is on top of the square and the triangle is in its original orientation, with A matched up with D and C matched up with E. Now how far has point A traveled altogether?

More

### Extra

Extra: If we used a pentagon instead of a square and did the same thing, how many rolls of the triangle would it take until A returned to its starting point on the pentagon? How about with a hexagon? Septagon?

The length of the arc is about 7pi/2 units, which is about 11 units.

• did you try using an actual triangle and square to model what happens?

• did you find that A doesn't move every time the triangle is rotated?

• and your answer is close, did you use a calculator early on and maybe round too much?