Paper Patterns for Building Cones

Date: 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/