I am a new knitter with very little training. I have yarn recommending size 8 (5 mm) needles, but my only needles are circular 16"size 4 (3.75 mm), how do I figure out how many stitches are needed for a hat fitting a 22" adult head?
If you’re a new knitter you might want to find a pattern that will work. Also with a 16 inch circular you’ll need to have DPN (double pointed needles) or another circular to finish the top because you can’t do it with a 16" circ.
What yarn are you planning on using? Name, weight and gauge please.
Usually, if a yarn company recommends a #8 needle, using a #4 on it will give you really, really tight gauge and stiff fabric. If you don’t mind that, you can knit a swatch–that is, cast on maybe 20 stitches–and see how many stitches make 4", then do the math from there.
It’s better, if you can do it, to buy another needle. A #8 is a very useful size for worsted-weight yarn, which is very common and probably what you have. The #4 is more useful for baby or light sport yarn. Jan is right about your needing another needle or double points to finish the top of your hat, unless you make one that doesn’t have a rounded top or has only a few decreases and is open at the top like the “messy bun” or “ponytail” hats. Once you’ve knitted for a while, you’ll be able to improvise, but for now it would be easier if you either changed the needle or changed the yarn.