I find myself needing to knit a disc. I want to join two cylinders with different radii.

What I need to knit is the perpendicular plane where they join. I actually need to make a ring rather than a circle, but the same maths should apply.

The equation to get the number of stitches to decrease each row would seem to be

where n is the number of sts to increase or decrease per row, g_sts is the st gauge and g_rows is the row gauge. What’s really neat about this is it’s independent of start and end radius altogether. Sts will need to be distributed evenly across the row and staggered throughout all rows so as not to distort locally.

So here’s a recipe for knitting up a circle for a given gauge and radius. It’s hardly a good way to go about putting less effort into knitting up a tension swatch as there’s a fair bit of maths to do first, but it does give a perfectly jaggy curve :)

Apologies for the eyeball-searing colour. I think I’ve been half-blinded from the real thing. I’m using Katia Monaco which is 100% mercerised cotton in a DK weight. The ball band claims a tension of 22 sts x 27 rows over 10cm on 3.5mm needles.

So this is the equation to get the stitch count for a chosen row (n) given the radius and the gauge. This will give you a quarter circle. To get a semicircle, double the value.

Which reads

“The number of stitches on row n is the square root of the radius (measured in stitches) squared, minus (the radius (measured in rows) less this row number) squared, times the gauge squared.”

A little cumbersome, perhaps, but all you have to do is plug in the numbers. With this yarn, for example, I wanted to get a 7″ Φ circle (17.7cm) and or 0.81.

So the radius is 8.8cm which means

And

So to calculate the appropriate number of stitches for each row, I made a table. I rounded the values to the nearest integers. I doubt even Elizabeth Zimmerma’am could knit 14.36 stitches!

The cast-on value is different to the number of calculated stitches because I adjusted for only casting on extra stitches at the start of each row. Note this is stockingette, so every even row is purled. Once you get to the end, work back down the rows, casting off instead of casting on stitches till you get to the last row and cast off the remaining 16.