Cut the cone along the line formed by the vertical segment of the string, and open it onto a flat surface. You obtain a triangle with a rounded base. The string slips off the cone when the angle at the top of the opened surface of the cone reaches 180 degrees. That is, when the opened cone looks like a semicircle.

If we take the radius R of the semicircle to be 1, the circumference of the cone base is given by:

C = 2*pi*R / 2 = pi

The radius of the cone base is then given by

r = C/(2*pi) = 1/2

The aperture α of the cone is easily calsulated as follows:

α = 2*arcsin(r/R) = 2*arcsin(1/2) = 60 degrees.

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