Ant's shortest distance that it can travel

Can you show me the answer in the cone unfolded. I would be glad if you guys can show me the solution in a simple way because I am a high school student.


Cut the cone along the segment $AT$. Unfolding, you get a sector of a circle. The apex $T$ of the cone becomes the center of the circle, and the point $A$ appears twice at opposite ends of the sector. The ant needs to go from one $A$ to the other $A$. Hints:

  1. Determine the radius of the circle.

  2. What fraction of the full circle does this sector represent? Hint: The base of the cone, when unwrapped, lives on the circumference of this circle.

  3. Now you know the shape of the circular sector, in particular the central angle of the sector. The shortest distance from one $A$ to the other $A$ will be a straight line.

| cite | improve this answer | |
  • $\begingroup$ Yes, I know that it would be straight line but when you fold it the ant is actually goes to the top and goes down again. There is no surface area that the ant travels. $\endgroup$ – CopperInTheSun May 30 at 0:46
  • $\begingroup$ Yes, unfortunately the shape of the cone leads to a degenerate solution: the shortest distance has the ant going to the apex of the cone and then back down. Any other path that reaches the 'far' side of the cone travels a longer distance than this. $\endgroup$ – grand_chat May 30 at 1:39

Not the answer you're looking for? Browse other questions tagged or ask your own question.