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Whenever I start a new Hat design, I invariably start with the crown. Even if I know exactly which style I’m aiming for or which stitch pattern I’m going to use, and whether it be a top-down, bottom-up or even sideways design, the crown is the first thing I consider. The maths of the crown determines not only the type of crown finish (beanie, beret, pixie, gathered or anything in between) but also the mathematical structure that runs all the way through a Hat.

#### THE ‘DECREASE 8STS EVERY OTHER ROUND’ FORMULA

This is a common formula for a Hat, and what it results in is a beanie style (half-dome, watch-cap) style Hat. However, it’s not applicable to every situation. Let’s look at some numbers.

With negative ease, this method works, as the edge around the top of the cylinder is smoothed off when worn.

The common formula for a circle is to decrease 8sts every other round, or 4sts average per round. This works well in something like stocking stitch, where the row and stitch gauge have a common relationship, or in lace, whereby blocking helps all the proportions settle.

#### THE ROLE OF THE CIRCUMFERENCE AND RADIUS

We’ll remember from school that the circumference of a circle is 2 x pi x the radius, where pi can be rounded to 3.14. In short, the circumference is 6.28 x the radius.

The circumference of our Hat is the number of stitches on our needles right before we start the crown shaping – it’s the widest part of the knitting, and the part that determines fit in a beanie. To know this measurement we’ll also want to know our gauge.

Let’s use an example of a Hat knit in DK with a gauge of 22sts/10cm. With a circumference of 100sts it will measure 45.5cm – the perfect finished size for the average adult female (allowing for 6.5cm of negative ease).

If we want to knit a beanie, and have the top of that Hat flat as a circle, we’ll be able to calculate the radius of our circle, which in turn helps us calculate the number of rounds over which we will decrease.

**45.5cm/6.28 (2 x pi) = 7.25cm crown circle radius**

If our row gauge follows a typical fashion of 3:4, our row gauge would be approximately 30rows/rounds to 10cm, or 3rows to 1cm, and so we would have to work our decreases over 22 rounds:

**7.25 (length of radius in cm) x 3 (number or rows/rounds per cm) = 21.75 rounds**

We have 100sts we wish to decrease, so we would aim to decrease 4.5sts per round on average. (100/21.75)

Except, that as we close the top of the Hat we don’t want to decrease down beyond say 6 or 8 sts, as this creates an unfortunate bump at the very top of the Hat, so we would change our decrease ratio slightly:

**(100-6)/21.75 = 4.32 sts decreased per round, average.**

Knitting is forgiving enough to allow us to decrease on average 4sts per round, and it would usually even out with blocking.

And don’t forget that that 4sts is an average figure – you could decrease 8sts every other round, or 12sts every third round and so on – it allows for quite a bit of flexibility when you need to adapt stitch patterns.

#### TAKING THE NUMBERS BEYOND STOCKING STITCH

But what if your row gauge and stitch gauge differ wildly, or your stitch pattern gauge doesn’t match that of your stocking stitch?

A good example would be a fairisle Hat where the row and stitch gauge are almost square, e.g. 22sts x 24rows, and the numbers would look a little different. Assuming we’re still working with an example of 100sts cast-on:

**45.5cm/6.28 (2 x pi) = 7.25cm crown circle radius**

Our row gauge gives us 2.4rows per 10cm, so we would need to decrease over 17.4 rounds:

**7.25 (length of radius in cm) x 2.4 (number or rows/rounds per cm) = 17.4 rounds**

We have 100sts we wish to decrease down to 6sts, the numbers would look like this:

**(100-6)/17.4 = 5.5sts decreased per round or 11sts decreased every other round.**

Another example might be garter stitch, where the row gauge is invariably double that of the stitch gauge, e.g. 22sts x 44rows to 10cm

**45.5cm/6.28 (2 x pi) = 7.25cm crown circle radius**

Our row gauge gives us 4.4rows per 10cm, so we would need to decrease over 31.9 rounds:

7.**25 (length of radius in cm) x 4.4 (number or rows/rounds per cm) = 31.9 rounds**

We have 100sts we wish to decrease down to 6sts, the numbers would look like this:

(1**00-6)/31.9 = 3sts decreased per round or 6sts decreased every other round.**

#### BEANIES vs BERETS

A beret essentially is the same idea as a beanie, except it’s a wider circle on top of a wider cylinder.

The same methods and formulas would be applied; first calculate the circumference measurement using the stitch count (at the point where you would start the crown shaping) and the stitch gauge. From there, calculate the radius and use the row gauge to determine how many stitches on average would be decreased per row/round.

As the maths of the circle is consistent, the number of stitches decreased per row/round should stay the same (4.32sts/row on average in Stocking Stitch) yet in practice, I have found that anything up to 6sts per round works well, too, even for stocking stitch. This is largely due to the larger area and the ability to block out smoothly.

#### TURNING A CIRCLE INTO A CONE

Having determined how many stitches on average we need to decrease across a set number of rows to make a crown flat, we can now think about what happens if we reduce or increase the decrease ratio (i.e. decrease more or less stitches than the beanie examples gave us)

If we increase the number of stitches we decrease, i.e. work less rows and on average decrease more per row/round, then we will head towards a gathered finish as the radius of the circle is rapidly shortened. If we reduce the number of stitches we decrease on average, and work more rows/rounds into the crown, then the radius lengthens, and creates a cone – this is how we create a pixie Hat.With a pixie Hat, the number of rows/rounds we decrease over is increased, and what was the radius of the circle now becomes the slant height (or angled edge) of the cone.

The circumference of the Hat has the same properties, but as a cone is essentially an incomplete circle, the circumference of the Hat would equate to the arc of the cone (if we were considering this cone as a flat construct)

The slant height of the cone is an important factor, as that determines the overall length of the Hat from the cast on edge. The actual height of the cone tells us how high the Hat is through the centre, and that requires a bit more maths. However to simplify things, you can simply add more non-decrease rows/rounds to the numbers that a beanie formula gives you – the more plain rounds you add, the taller your Hat will be.

One final thing to bear in mind when designing pixie Hats – the length of the body doesn’t need to be as long, as head will sit higher in the crown. You will still get that lovely pixie finish, but it is better supported (i.e. it will stand up straight rather than flop over!) if the crown shaping is started a little sooner.

#### THE GATHERED FINISH

The gathered slouchy look has been popular of late, and it follows a similar formula to other shapes of this nature, in that you’ll knit a cylinder for the body. To create that gathered look we want to decrease more rapidly, and exaggeratedly so. When I design slouchy Hats with this type of crown I generally aim for decreasing either half the stitches per decrease round with only a few plain rounds in between, or an average of 10sts per round.

However, simply gathering all the stitches up at the end of a cylinder and finishing there isn’t advised – this creates quite a bulky and unflattering look at the crown, hence a gradual but rapid decrease formula is better. Similarly, a plain round before threading your yarn through the stitches and pulling to close is always advised, as it soften that final part of the Hat.

#### GOING BEYOND THE CYLINDER + TOP STRUCTURE

It’s worth bearing in mind that this structure is only one way to construct a Hat; there are countless other ways of doing it. That said, it’s a good place to start, so that you understand the maths and numbers which will then enable you you to go beyond this method.

So often I see knitters and designers blindly following the 4sts every round/8sts every other round formula without fully understanding why those numbers fit, and their limitations.

this article was originally published in issue 97 of the Knitter, Spring 2016

There’s a lot more involved in designing a Hat than I’ve shared here! If you’re interested in learning more, I’ve a number of self-paced, downloadable online classes all about Hat design, and you can find them all in my Hand Knit Hat Academy!

You are amazing! Truly the professor of hat architecture! The makes so much sense and explains why some of my hats are too short, which is maddening!

But I need to ask, what hat has the gorgeous crown pictured at the top of the article? It is lovely! Reminds me of cathedrals!

That’s the crown shaping for my Aeonium pattern – available either as a single or as part of Twisted Woolly Toppers 🙂

http://www.woollywormhead.com/aeonium

Master class indeed. And somewhat beyond my math skills. So I will leave it all to WW and continue to buy her patterns!

Thank you for explaining your process . I will probably never design a hat but it’s always good to understand why and how a specific design was accomplished. I know with your hats , you’ve done the math so I don’t have to.

My head just fell off and rolled down the hall….too smart for me!! Good thing we only have to follow your patterns 🙂 You’re amazing.

What?? Never mind. I will buy and follow your patterns and happily knit them to beautiful! (you’re pretty darn awesome)

Ow. My head hurts. I’ll have to study that more at another time. It is fascinating.

I really appreciate your efforts, but I am one of those who are not "math inclined" period… So I will continue to rely on your wonderfully written pattern instructions as I have always done. You’ll never have to worry about me competing in the hat design business.

I really enjoyed reading through this and digesting each section. It really makes sense now why you’ve been able to create so many different styles. This breakdown speaks volumes for your skills. Thank you for sharing!

Thank you for this very clear and helpful mathematical explanation. I found myself redesigning a poorly arranged fair isle hat, using a very different weight of yarn, and thus a different stitch count, from the original design, and realized that I could not follow the original directions for the crown shaping. Your explanation has helped me out of the impasse and set me up to finish the hat with success!

oh wonderful, I’m so glad you found it helpful!

I am going to be coming back to this often. Thank you so much for this post. Also, how do we find the workshop? LYS isn’t doing much right now, do you have an online course? Love that hat crown! So beautiful.

Thank you, I hope this article helps you!

I’m developing online workshops at the moment, with the first 2 ready to go live in this month (February 2021), all being well. The crown shaping masterclass hasn’t been recorded yet but the first classes, including one free one, all cover different aspects of Hat design. My newsletter is the best place to be to hear of new pattern, eBook and workshop releases 🙂

Wonderful explainer and very helpful. Now I will print it and convert all the centimeters to inches since I am old and my brain only thinks in inches…

Thank you. I didn’t know why some hat patterns knit up beautifully & others not so much.

I now know why, it’s all in the planning.

I like crown decreases that continue the pattern of the hat.

Yes, planning makes such a difference – the crown shouldn’t be an after thought!

Continuation of the Hat patterning from the body into the crown is something that makes me happy, too 🙂