Sock Math

Most knitted sock patterns follow the same formula, because they all work with the same proportions. One of the things I like about knitting socks is that the math is so predictable and easy.  Once you have the number of stitches to cast on, you have almost all the numbers you need to make the whole sock.  Knowing the numbers behind sock knitting will also help you figure out what's wrong if you are struggling with following a sock pattern. 

Here's how it works:

  1. The first thing you need to make socks is the circumference of your foot (or that of the foot for which you are knitting).  That measurement is similar to your ankle measurement. You also need your stitch gauge in stockinette. Socks are made using a gauge that's a bit tight compared to what you would normally get with your chosen yarn.  Socks made in worsted (like the sample pictured) are often worked on US size 5 or 6 needles (I used 5s).  DK yarn would be worked on 3s, and sock yarn (fingering) on 0s or 1s, instead of 2s or 3s. 
  2. Multiply your foot circumference by your stitch gauge, and then round the result to the nearest multiple of 8.  That's the number of stitches you cast on. If you look at most sock patterns, they start with a multiple of 4 stitches. That's not because of the pattern stitch used on the cuff, but because of how you use the stitches when you get to shaping the sock. Multiples of 8 work best, and multiples of 4 are the easiest to finagle if you can't start with a multiple of 8. For the sake of easy numbers, I cast on 48 stitches. 
  3. Make your cuff.  Sock cuffs can be whatever length you prefer. That said, the cuffs on crew socks are about the same length as the foot, ankle socks have cuffs half that length, and no-shows have about an inch of cuff.  For the sample pictured, I worked about 2 inches of k2, p2 rib. 
  4. Once the cuff is done, you move on to the heel flap.  The heel flap is worked in rows across half your stitches, and it is a square.  In order to ensure a square flap and make it durable, heel flaps are often made by working (sl1, k1) across the knit side and then purling back (shown below). You will work as many rows as you have stitches on the heel flap, and you slip the first stitch of each row.  Slipping the first stitch is important, because those slipped stitches are where you will pick up the gusset stitches later. Since I started with 48 stitches, I have a 24-stitch heel flap, worked 24 rows, and slipping the first stich of each row gives me 12 slipped stitches on each side.  
  5. After making the heel flap, you work short rows to turn the heel.  Starting with a right side row, this is done by working halfway across the flap, knitting one more stitch, knitting 2 together, and knitting 1 more. Then you turn, purl 4, purl 2 together, purl 1 more, and turn.  Now you have the center two stitches with a decrease on either side and a single stitch beyond each decrease.  With each successive row, you work across to the decrease, and then knit or purl 2 together using the stitch after the decrease and picking up the next, previously unused, stitch on the needle. When you don't have any more stitches to pick up, you make sure you have ended with a right side row, and now you are ready to pick up the gusset stitches and work in the round again. Turning the heel should give you half the number of stitches from the heel flap, plus 2.  However, this isn't rocket science. If you end up with a turned heel and an even number of stitches, it's good enough. If you end up with an odd number of stitches, either make an extra decrease in the final row turning the heal, or add an extra decrease on one of the gussets. 
  6. Note: I'm going with the hypothetical scenario of using a set of four dpns, which is what I prefer.  I keep half the stitches (the instep) on one needle, and a quarter of the stitches on each of the other two.  Most people use a set of five dpns, with the instep stitches divided between two needles.  I'm going to describe making the foot using a set of four needles simply because I find it easier and less confusing to describe. Divide your heel stitches evenly between two needles.  Using the needle where the working thread is attached, you pick up a stitch in each slipped stitch down the side of the heel flap. This is now Needle 1, and the beginning of your round. On the next needle, work across the instep stitches (the stitches from the cuff that you haven't been using).  That's your new Needle 2. Then pick up a stitch in each slipped heel flap stitch on the other side of the heel flap and knit across the remaining heel flap stitches with the new Needle 3 to get to the end of the round.  
  7. The number of stitches you picked up along either side of the heel flap should be equal to half the number you cast on.  Going with the hypothetical cast on of 48, you would have picked up 12 stitches on each side of the heel flap.  That means that you now have 48 stitches plus the stitches from turning the heel. I had 14 stitches left from turning the heel, so my total stitches at this point were 62.
  8. Now you are ready to make the gusset. Every other round, you work to the last 3 stitches of Needle 1, k2tog, k1, work across Needle 2, k1 ssk, and then work to the end of Needle 3.  You work the round between decrease rounds even.  The gusset is done when you have decreased back down to the original number you cast on. Like I said before, this isn't rocket science; it doesn't have to be perfect.  If you have an odd number of stitches at this point, you can either work one more decrease on one of the gussets or you can wait and do that extra decrease in the toe shaping. Neither option is going to make a significant difference in the appearance of the finished sock. The pattern stitch you use on the instep stitches from here on is not important.  For the sample pictured, I continued to use k2, p2 rib down to the beginning of the toe shaping.  The diagonal line that runs away from the heel flap and off to the left is the gusset, and it's made by making sure my decreases lined up.
  9. Once the gusset is done, you just work even, without increasing or decreasing until the foot of the sock is almost the length of the foot it's supposed to fit. For a big sock, that's going to be an inch or so from the desired finished length.  For a baby sock, it's more like half an inch from the finished length.  Your row gauge also matters here. That said, the number of rounds in the toe should be about one quarter of you number of stitches. Whatever measurement that is in your row gauge is how much space it will take up.  Using my sample cast on of 48, I decreased down to 24 stitches at a rate of 4 stitches every other round.  That means 6 decrease rounds, with a round worked even between each decrease round, so 12 rounds of toe shaping.  If I'm using thick yarn and have 6 rounds per inch, then I start my toe shaping 2 inches short of the finished foot length.  If I'm using thin yarn and have 12 rounds per inch, then I need 1 inch for the toe shaping. 
  10. Decreasing for the toe involves decreasing 4 stitches every other round until you have half the number of stitches you started with. The decreases occur along the sides of the foot, so you work across Needle 1 until 3 stitches remain, k2tog, k1. On Needle 2, you k1, ssk, work across until 3 stitches remain, k2tog, k1. In most sock patterns, you stop using any pattern stitch on Needle 2 at this point and work the toe in straight stockinette. On Needle 3, k1, ssk, work to end of needle. If you started the sock with 48 stitches, you decrease until you have 24 stitches around.  At this point, it's important that you have an even number of stitches.
  11. Once you are done decreasing for the toe, you should have half your stitches on the top of the foot and half on the bottom.  You cut the yarn with a good length tail, and then sew the toe shut using kitchener stitch. If kitchener stitch seems scary (It's not, but it can feel a little overwhelming at first!), you can also close the toe using a three-needle bind off, but that will leave a visible seam on your sock toe that some people find uncomfortable on their feet.
And that's how a traditional knitted sock works! It doesn't matter what pattern stitches you use or what yarn or needles you've chosen, these proportions are pretty universal.

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