My gram used to make those stools also. And I thought it was her idea all these years. I remember them and loved them. I started one years ago and never finished it.
I was taught by my husband's grandmother the technique you are looking for: you will need 7 big juice cans with the lids still intact (use a bottle opener to simply put 2 holes in the top to empty them); 7 old socks; 1 yard of material (polished cotton works well) 45 in. wide; thin fishing line for your 'thread'; Polyfill/Cotton Filling for top; hard cardboard or foamboard for top and bottom of stool. Cover the 7 empty cans with old socks. Cut from your material 6 pieces of fabric 12x12 and 2 pieces from the remaining material - these two are for top and bottom of stool. Cover the 6 cans with the 12x12 pieces, pin in place and sew up the long side of the cans. Now arrange them around the remaining canin in place to hold and sew all the cans together at the sides, tops and bottoms (make sure your seam side is inside the stool towards the center can before you start to sew! Cut your cardboard for the top and bottom of the stool after tracing around the sewn together cans. Put some old socks or whatever around that center can so it doesn't jiggle. Now pin in place the bottom fabric and then sew it on the almost finished stool. Do the same to the top but leave some room for the filling. You will need to keep tucking the extra fabric between the cardboard and the cans and sewing, tuck and sew, maybe even cutting fabric into points because you're working with square fabric and round cans. Once you have the top of the stool about 1/2 stitched on, start filling it with the polyfill and pushing it in the round corners with a ruler or dowel rod, then keep sewing. Do this until you're finished. I always use piping or cording around the top and bottom edges to hide my stiches and give it a finished look. Good luck!!
My grandma made many of these stools. Most of the grandkids were able to get at least one before she passed away. She would also stuff on the outside of the cans so you ended up with a round outside and so could not tell that there were juice cans inside the stool. She padded the top also. She bought remnants at the local MCC store (thrift store) and was given much material also to use and so the stool was very reasonable in cost.
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