Paper Patterns for Building ConesDate: 11/03/2003 at 16:23:29 From: mike Subject: how do I make a pattern for building a cone I am building a cone shaped fish trap for teaching kids about fish living in the salt marsh. The mouth will be 26" in diameter and the overall length will be 72". Is there a way to make a paper pattern? Date: 11/04/2003 at 06:04:10 From: Doctor Jeremiah Subject: Re: how do I make a pattern for building a cone Hi Mike, You mentioned that 72 inches is the distance from the opening to the point on the end, and 26 inches is the diameter of the oepn end. If we look at the rolled-up cone from the side: |---------72---------| | | --+-------- + | | | + | | | + | | | + | 13 | + | | | + | | | + | --+-------- | + | | + \ | | + \ 13 | + | | + | | + | | + side length (L) --+-------- + \ \ First we need to find L. Looking at half the cone, we see a right triangle: |---------72---------| | | --+-------- +--------------------+ | | + \ | | + \ 13 | + | | + | | + | | + side length --+-------- + \ \ We can calculate the side length using the Pythagorean Theorem: side_length^2 = 13^2 + 72^2 side_length = square_root(13^2 + 72^2) side_length = 73.164 inches So draw a circle that has a radius of 73.164 inches. +++++ ++++ +++ \ +++ \ + \ + \ + \ \ + \ \ + Pi*26 inches around outside \ + | \ + } +-------------------+ --- |---73.164 inches---| We need to know where to cut the large circle to make the cone shape. We know the open end of the cone is a circle with diameter of 26", so the circumference of that circle is Pi*26, which is equal to 81.68 inches. If you were to cut a string that long and measure around your big circle, you would know how much needed to be cut out. But don't cut yet! You also need to add an overlap so that you can glue the thing together. The overlap doesn't get smaller as you go toward the center, so it would look like this: \ +++++ 2 ++++ +++ \ / \ +++ \ \ + \ \ + \ \ + \ \ \ + \ \ \ + 81.68 inches around \ \ + | \ \ + } \---+-------------------+ --- | | |---73.164 inches---| Measuring around with a string may seem silly, but it is the easiest way to measure it. If you want a more mathematical answer, then in percent it is the distance around the outside (81.68 inches) divided by the total circumference of the large circle (2*Pi*73.164 inches = 459.7 inches) or: % of circle = 81.68 / 459.7 = 17.768 percent. Since circles have 360 degrees, our cone will have 17.768 percent of that or: 360 * .17768 = 63.97 degrees You could measure that with a protractor, but it would need to be a really big protractor or else it wouldn't be very accurate. The string idea might even be more accurate, especially if you are very careful. Note that I made my drawing before calculating the angle, so it looks odd to have an obtuse angle in the drawing when the measure of the angle is in fact about 64 degrees: \ +++++ 2 ++++ +++ \ / \ +++ \ \ + \ \ + \ \ + \ \ \ + \ \ \ + 83.68 inches (including overlap) \ \ + | \ \ 63.97 degrees + } \---+-------------------+ --- | | |---73.164 inches---| - Doctor Jeremiah, The Math Forum http://mathforum.org/dr.math/ |
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