Triangular Wheel
An equilateral triangle is rolled along a line. What is the length of the path traced out by one of its vertices?
Age
14 to 16
Challenge level
Problem
Image
The equilateral triangle ABC has sides of length 1 unit and AB lies on the line XY. The triangle is rotated clockwise around B until BC lies on the line XY. It is then rotated similarly around C and then about A as shown in the diagram.
What is the length of the path traced out by point C during this sequence of rotations?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
In each rotation which C makes, the radius of the arc it describes is 1 unit. In the first rotation, C turns through an angle of $120 °$, so it moves a distance $\frac{1}{3} \times 2 \times \pi \times 1$, that is $\frac{2\pi}{3}$.
As it is the centre of the second rotation, C does not move during it.
In the third rotation, C again turns through an angle of 120 °, so the total distance travelled is $2 \times \frac{2\pi}{3} = \frac{4\pi}{3}$.