A right circular cone has radius r and slant height s.
A fly starts at a point on the edge of the cone's base,
walks around the vertex, and returns to its starting point.

What is the minimum distance traveled in terms of r and s?

When the surface of the cone is laid out flat it forms a sector of a circle. The arc length of this sector is 2*pi*r, and its radius is s. The arc constitutes, therefore, (2*pi*r)/(2*pi*s) of the full circle and therefore subtends an angle of 2*pi*r/s radians.

The path of the fly is the base of the triangle formed by the vertex of the sector and the end points of the arc (i.e., it's the distance between the endpoints of the arc). This isosceles triangle can be broken into two right triangles with the right angles meeting at the midpoint of the fly's path. That path therefore has length 2*s*sin(2*pi*r/(2*s)) = 2*s*sin(pi*r/s).

The above assumes that r/s < 1/2. Beyond that, the circular sector of the surface laid flat will be concave at its vertex, and the distance found will be cutting across non-surface. When r/s > 1/2, the shortest route for the fly would be to head directly for the vertex of the cone, do an infinitesimal loop around it and head straight back to the start for a total distance of 2*s.

Posted by Charlie on 2009-07-01 13:46:23