[MODNOTE: Also see post #24 for a spreadsheet by bionut which does some of these calcualtions for you.]
This is a step-by-step tutorial for generating a fabric pattern for the two main pieces of fabric you will need to sew a fustrum(truncated cone)-shaped brew bag. You will cut one circular panel that will be the bottom of your bag, and one long, curved panel that will form the wall of the bag. I’ll be giving examples of the math involved using numbers from my brew pot, as well as the formulas you’ll use to identify the dimensions you’ll need for your brew pot. The dimensions I’m using will be in inches, but the formulas will work using any unit of measure you like, provided you use the same unit of measure for all the variables. This method also assumes you have a (roughly) cylindrical brew kettle. I will also document the process of sewing the bag, however there are many different techniques so the one represented here is just my own and may not be the best one, or the best one for you.
Step 1: determine the height (h), upper radius(r1), and lower radius(r2) of your fustrum.
Measure the height of your brew kettle and multiply that number by 1.15. Then add an allowance for your drawstring/webbing/tabs/handles/whatever you want to add to the top of your bag. This value will be “h”, and represents the height of the fustrum. My brew pot is 15.25” tall, and I want an extra 1” on the top of the bag for nylon webbing and handles. So:
Formula: (brew kettle height x 1.15) + handle allowance = h
Example: (15.25” x 1.15) + 1” = 17.40”
Measure the outer diameter of the top of your brew kettle and multiply that number by 1.15. Then divide that number by two, to get the upper radius (r1) of your fustrum.
Formula: (brew kettle upper diameter x 1.15)/2 = r1
Example: (12.75” x 1.15) / 2 = 7.33”
Measure (or estimate) the inner diameter of the bottom of your brew kettle, and multiply that number by .85. Then divide that number by two, to get the lower radius (r2) of your fustrum.
Formula: (brew kettle lower diameter x .85)/2 = r2
Example: (12.25” x .85) / 2 = 5.20”
Step 2: determine the total height (ht) of the cone that your fustrum is a part of.
Since the fustrum is a cone with the top removed, the total height of that cone without that top removed is essential to generating your fabric pattern. That number can be found using the following formula:
Formula: (h x r1) / (r1 – r2) = ht
Example: (17.4” x 7.33”) / (5.20” – 7.33”) = 59.87”
Step 3: determine the length of the side of the cone your fustrum belongs to (b) and the length of the side of the cone removed to make your fustrum (a).
Referring to figure 1, you can see what the dimensions of “a” and “b” represent. These figures are essential to creating the fabric pattern of your bag, as they describe the radii of the two arcs you will draw when putting your pattern on the fabric (refer to figure 2). These are most easily determined using the right triangle calculator at http://www.csgnetwork.com/righttricalc.html" onclick="window.open(this.href);return false;.
Directions:
Input (ht – h) as “Side A”, r2 as “Side B”, and calculate. The value in “Side C” will be value (a) on figure 2.
Clear the values, and then input ht as “Side A”, r1 as “Side B”, and calculate. “Side C” will now be value (b) on figure 2.
Example:
My ht is 59.87”, and my h is 17.40”. 59.87” – 17.40” = 42.41”, so I input this as “Side A”. My r2 is 5.20”, so I input this as “Side B”. The calculator gives me the result of 42.73” as my value for a.
I then clear the calculator and input my ht of 59.87” as “Side A”, input my r1 of 7.33” as “Side B”, and the calculator gives me the result of 60.32” as my value for b.
Step 4: determine angle c
When transferring this pattern to fabric, you will begin drawing an arc at a fixed radius, and then continue that arc until it’s as long as r1 or r2, depending on the radius of your arc. Instead of trying to measure distance along a curve ,the easiest way to do this is to determine the angle at which that arc is the distance you want. If your radius calculations are correct, you will achieve the correct r1 and r2 distances at the same angle. The method for that is as follows:
Formula: (r1 x 2 x 3.14) / b = Angle c
Example: (7.33” x 2 x 3.14) / 60.32” = .76 radians
As you can see, this formula gives the angle in radians, which you must then convert to degrees by multiplying that value by 57.3.
Formula: radians x 57.32 = degrees
Example: .77 x 57.32 = 43.56 degrees
So, in my example Angle c is equal to 43.56 degrees.
Step 5: draw your pattern on the fabric
In steps 3 and 4, you have calculated all the values you need to draw your pattern on the fabric. Referencing Figure 2 you will see that two arcs are necessary; one with a radius of “a” and a distance of angle c, and one with a radius of “b” and a distance of angle c. The easiest way to do this is to use a protractor to mark an angle on the floor that is equal to angle c. Then, using a measuring tape or string, measure out the distance of “b” from the apex of the angle, aligning the string or tape with one side of the angle you marked. Now, pivot that string or tape around the apex of the angle, marking along your fabric at the same distance from the apex, until your string is aligned with the other side of the angle. Repeat this process, using radius “b”. Connect the ends of these arcs together, and you have the first panel of your pattern, the wall of your bag!
You can double-check your work by measuring along arc 1 to approximate its length. It should be close to the diameter of your pot, multiplied by 1.15, multiplied by 3.14 (D x 1.15 x 3.14). Check the height of your bag by measuring along “d” (figure 2). It should be slightly larger than the height of your brew pot multiplied by 1.15 (H x 1.15).
The second pattern is easy: mark a circle with a radius of r2 . That’s it!
Keep in mind, these patterns do not have seam allowances, so you’ll probably want to cut ½” outside of the pattern to give yourself a little extra fabric to work with. Once the panels are cut out, sew your circle to the edge of arc 2, and connect the ends of the wall panel of your bag. That’s it, you now have an inverted fustrum brewing bag that’s custom sized to your brew kettle. All it took was a little math!
I'm going to sew this thing up tomorrow (hopefully) and I'll try to take a bunch of pictures to make this a little more practical.
instructions for sewing a truncated cone bag
Post #1 made 11 years ago
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Last edited by mr.jadkowski on 26 Mar 2013, 03:38, edited 2 times in total.
. You've put a lot of time and effort into that post #1 - very nice 
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