Quandaries
and Queries |
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sir i need to make a conical hat for my daughter's upcoming fancy dress, the circumference of the base(hollow) is 50 cms,the height of the cone is 30 cms,what should be the dimensions of the paper which will make a cone of the beforementioned dimensions? Regards, Manish |
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Manish, If you had the hat and cut it straight up from the base to the peak and rolled it out flat you would have a sector of a circle.
The relationship between the radius r, the length of the arc a and the angle at the center theta is
where the angle is measured in radians. You have a = 50 centimeters and r = 30 centimeters. Hence
which is
My suggestion is that you make a paper model of the hat before you make the real one. I would make the centre angle a little larger than 95.7 degrees so that you have some overlap when you roll it into a cone. Penny We received a reply,
Oops, I misread your question. To answer your question correctly I need to draw a triangle ABC inside the cone with A at the apex of the cone and B at the centre of the base.
The length of AB is 30 cm and the length of BC is the radius of the circle. The circumference of the circle is 50 cm and the relationship between the circumference and the radius is
Thus
and hence the radius is The triangle ABC is a right triangle so by Pythagoras Theorem
Thus
and hence the slant height |AC| is 30.1 cm. Now I can go back to the second diagram in my first answer and change the slant height to 31.0 cm and continue as before
The relationship between the radius r, the length of the arc a and the angle at the center theta is
where the angle is measured in radians. You have a = 50 centimeters and r = 30.1 centimeters. Hence
which is
Penny |