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EDIT1:

Changing description on how to bend paper of $A4$ size for example to make portion of a cone

The long side $L$ of a rectangular paper sheet $L \times W $ is bisected and without any cutting the two half length $L/2$ edges are rolled out of plane/ bent/glued to form a pointed vertex of a right circular cone of semi vertical angle $ \alpha$. Such bending creates parts of a cone with two discontinuous edges:

  • one edge of combined length $2 W$ double width boundary brought into alignment

  • one bent length $L$ of main boundary

  • ( one merged / glued/slightly overlapped generator of length $L/2$),

Bending has preserved lengths and angles. Find $ \alpha.$

Another way to make such a cone is as follows:

Fold a paper $ L \times W $ into two parts lengthwise of two double sheets of $ L/2 \times W $. Glue half lengths. Make the flat fold round as if to fill in some air inside, making a cone vertex. Find $\alpha ,$ the semi-vertical angle of the bent cone.

Narasimham
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  • "the two halves glued together after adjusting/ bending paper to form a right circular cone", so Is the area of the cone equal to the area of rectangle? Also, is the length of the cone equal to the long side of the rectangle? – Narek Margaryan Oct 27 '15 at 06:58
  • Is the edit ok? Bent rectangle has same area. Slant lengths change everywhere. – Narasimham Oct 27 '15 at 08:45

1 Answers1

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Had you cut the sheet as a half-circle of diameter $L$, you would have obtained a cone with the circumference of its base equal to $\dfrac{\pi L}2$, hence radius of the base $\dfrac L4$ and apothem $\dfrac L2$.