Out of a circular piece of paper, you wish to form a cone cup, so you cut out a circle wedge (with its extreme at the circle center) and join the resulting straight sides, forming a conical cup.

What size should the wedge be, to maximize the capacity of the cone?

Well, unless I've made a mistake here, the maximum capacity cone will be achieved by cutting out a wedge from the circle with a central angle of 2pi[1-(sqrt(6))/3] and discarding it.  The remaining piece of the circle can be folded into a cone which should have the maximum possible capacity.  This central angle of the to-be-discarded wedge is approximately 66deg3'40.43", using a hand calculator.

This result seems reasonable, intuitively... I wonder if it is correct?

-John

ps Had to edit this, as I can't get the numerical symbols to look right!

Edited on April 1, 2005, 8:20 pm

Edited on April 1, 2005, 8:22 pm

Posted by John Reid on 2005-04-01 18:41:45