What's (geometrically) wrong with my peg bag.

I should be cleaning the kitchen, but instead I bring you: what is wrong with my peg bag (geometry edition).

From this ugly photo of the ugly bag, I think part of the wrongness arises because the sides of the bag taper inwards towards the bottom.  I'm not talking about the round-bottom shaping, but the slightly sloping sides.  They're not sloping enough to be style, just enough to be looking like a mistake.  And that's because they are.

I put two darts in the bottom of the bag to give volume for pegs.  I failed to calculate that this would make the slides slope inwards and make the bottom of the bag narrower than the top (well, duh).  In the interests of making my pattern correct next time, I thought I'd calculate what I was supposed to do when I drew those darts on my pattern.

Towards the bottom of the bag, my pattern should look like the image above with darts of angle theta and side l, and dart points separated by a distance x.  The total width of this pattern piece is z.  I also marked distance d and delta, which come in handy later.

 

When I come to sew up the darts, the bag will appear to have a width w when viewed from above.  This is narrower than z marked on my pattern piece, because the darts have given the piece some three-dimensional shape.

When viewed in cross-section, lying on a flat surface, the middle of the fabric will rise above the surface by height h (at least, if it were made of stiff card, not floppy cotton).

The interesting quantities to me when I design the bag are theta and h.  The dart angle lets me choose how "boxy" I want the bag to be:  theta = 0 is no shaping at all, theta = 90 degrees is a square bottomed bag, and intermediate values give gentle shaping.  The quantity h allows me to choose how wide back-to-front my bag will be inside.

From my choice of theta and h, I can calculate some useful design parameters which help me to draw my darts in later.  Time to get out my calculator and calculate these numbers:

One can also find a few more useful numbers, in particular x and l which tell me the separation of my dart points, and how long the dart sides should be.

Finally, the total width of my pattern piece at the lower end should be z:

... while at the top by the hanger (where there are no darts), it should just be width w.  So, there we have my mistake, my friends: it's equal to z-w, if you cared.

I can now re-visit my wrong pattern with real numbers.  I'd just drawn darts in at random, but it looks like I used an angle of theta = 28 degrees, with a dart length of l = 4.5cm.  The top of my bag has width w = 34cm.  I find that tan(xi) = 0.601 (that's tan(squiggle), where "squiggle" is clearly the best greek letter), and  d = 3.86cm.  It means the inside of my bag has h = 3.1cm, but really it's 6.2cm across because I used darts in the back and front bag pieces.  Now for the interesting bit: the bottom of my flat pattern (near the darts) should have been z = 37cm across, not the 34cm that I used.  That's an 8% error: enough to make my bag look crap.

Time to re-draw!

I'm also going to change the mouth of the bag, add elastic loops inside to hold the hanger up, and a small fabric loop on the outside at the bottom to pass the hook through when you fold the bag in half for storage.

Sorry for the algebra explosion: if anyone thinks this sort of thing will actually be useful for them then drop me a comment and I could make a calculator-type thing, if maths isn't your forte...