Let's say your sister gives your mum some extra-wide sheets to somehow reuse. And you happen to be around before your mum starts anything with the sheets. I don't know about you, but that sounds like bath mat making time for me! Especially as I had hoped to make one on my holiday, anyhow, and therefore had my 15 mm circular with me.
I took the sheets, a baby's bath tub to contain them and a pair of scissors and sat on a garden chair in my mum's garden, and started cutting ca. 2 cm (bit less than an inch) wide strip from the fabric. I cut, and rolled the strip into a ball, cut some more, rolled some more. The cotton sheet was easy as I could rip it, but the frottee one was more difficult as it was knit and not woven, so I had to cut it from beginning to the end.
When it started to rain, I sat on the covered porch, cutting and rolling. It took me a few days to finish ripping both huge sheets as I obviously didn't just do that all day long. It must have been just a few hours of concentrated work to finish the balls which ended up the size of huge cabbages!
I first knit the frottee one, as I had much less "yarn" for that (it was a mattress sheet, as the other one was a duvet case). I cast on 22 stitches and knit every row until the yarn was gone.
Then I made the cotton one. I cast on 25 stitches and worked in 5-stitch basketweave for seven repeats (in short: (k5,p5) x 2, k5 for 10 rows, then (p5,k5) x 2, p5 for 10 rows, repeat both two more times, then work the first section once more). I still had some yarn left, but the mat was big enough so I bound off there.
The cotton mat is about twice the size of the frottee mat. I currently have the cotton mat in our bathroom; I'll put the frottee one there this autumn, I think. My knitted old bath mat is waiting to go into the dyeing pot to cover old stains. Let's see how that works out!