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?



    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?

    Answer Check

    Show Answer

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

    If your answer does not match our answer,

    • 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?

    If your answer does match ours,

    • have you carefully explained each step that you took?
    • did you make any mistakes along the way? If so, how did you find them?
    • did you try the extra?

    Teacher Materials