How to use trigonometry for greater accuracy when slashing and spreading

Today's post is a supplemental deep dive into rotation vs. slash and spread pattern drafting, specifically for drafting a hat brim.

Hello mathy-sewing friends! I’m really excited that I have figured out a solution to a drafting problem/query using math. It’s honestly the most thrilled I have felt in a long time. 😁 This feeling of elation at solving a problem is a big part of why I majored in math.

When I first started researching hat drafting, I was disappointed that there were not clearer instructions and explanations of why things were done this way or that way. As I’ve worked on the tutorials in this series, I’ve done a lot of experimentation—drafting digitally and with paper, making paper mockups, tweaking and repeating.

In today’s post, I will compare the two drafting methods and then show how you can use math to calculate a space interval, given an angle of rotation. (This is helpful if you are drafting on paper.) We will be using trigonometry and the sine function so grab your scientific calculators!

Click here to catch up on the previous posts in the hat drafting series.

Two weeks ago in the Bucket Hat tutorial, I talked about how the slash and spread method of using a set space interval creates a different curve depending on the brim depth. I’m re-sharing the comparison image I created above. You can see that the pattern pieces created using the rotation method nest together beautifully whereas those created by spacing have different curves and do not nest.

A Comparison of Methods

In the chart above, I recap the data of the brims shown in the first image of this post. Each brim is 11.375” wide and divided into 8 pieces. There are five different brim radii running from 2” to 3”.

For the first set (in blue section above), I rotated each segment 10 degrees. After rotating, I measured the space interval and the outer brim length*. The space interval varied from .3486” to .5229”, increasing as the brim radius increased. Likewise, the outer brim length increased with the brim radius.

*Note, when I measured the outer brim length, I measured the straight segments and did not draw a new curve. Drawing a new curve would change that measurement.

For the second set, I pivoted each segment to create a .5” space at the outer edge. In Illustrator, I was able to measure the degree of rotation for each adjustment. Keeping the space interval the same resulted in a decrease of degree of rotation as the brim radius increased. Technically, when measuring the straight segments, the outer brim radius should be 11.375 + .5*7 = 14.875 for each of the brims in this set. (I was a little bit off in my digital spacing but the result was very close to 14.9 for all brims. The margin of error looks to be about .02”, which I think is acceptable for this casual comparison.)

I also made scaled down mockups of three brims (2”, 2.5” and 3” brim radius) because I think it really helps to see things in 3D. (A lot of pattern making relies on testing things out. A 2D draft might look perfect but then when sewn together, one often finds areas that can be improved or even new insights. Personally, I often understand things better upon seeing them “in action”.)

Above are side by side photos. (Admittedly, I should have set up a tripod for more consistent angle of photo.)These brims are scaled down to 42% (for ease of printing). The brims on the left were all created with a 10 degree rotation. The ones on the right use a .5” spacer. You might be able to see that the angle of the brims on the right becomes steeper as the brim radius increases.

Sample brims stacked together

From the mockups, I do believe that the angle of the brim is more consistent with the rotation adjustment. Above are photos of each set of sample brims stacked together, with longest brims on the bottom. The rotated brims on the left fit together snugly with no gaps. The spaced brims on the right, stack but are a little wobbly. In the photo, you can see them tipping and gapping.

Overall though, it’s not a very significant difference. However, if you compared a 2” brim to a 6” brim, the angles would be significantly different.

Conclusion

So, the big question is, how important is this? I think it’s fun to look at the comparisons and analyze why different methods give different results. But, in the long run, this level of detail is probably not a big deal, especially if you are only making one size of hat.

Reasons to use a consistent angle of rotation

The main benefit of using an angle of rotation is that the angle of the brim is consistent. If you are drafting a multi-size pattern, using an angle of rotation to slash and spread would make a more consistent result across all sizes. And, as we can see from the first graphic in this post, the pattern pieces nest together very nicely.

For me personally, I have had hats where the brim angles too much. So, finding the ideal angle is important to me. It may not be as important to you and that’s ok.

Reasons to use a space interval

One possible benefit to making the adjustment by space interval is that the outer brim would be equal across sizes. This might be handy if you were attaching a trim or other pattern piece to the outer brim.

The second, maybe most important reason, is that if you are drafting with pen and paper, it’s easier to draft using a space interval. Drafting digitally I think it’s easier to use the rotate tool. But with paper, it’s hard to draw an exact 14 degree angle.

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A little bit of background before we get into the fun stuff, aka the math

When I wrote the tutorial for the bucket hat, I hadn’t yet figured out how to mathematically translate that degree of rotation into a space interval for paper drafting. I’m really excited that I figured it out last week. And we get to use trigonometry!

As I mentioned earlier, in Illustrator, it’s very easy pivot an object to a specific degree but it’s tricky on paper. It really bothered me that I wasn’t able to easily achieve the same results using paper and drafting digitally. So, playing around with paper drafting, I drew a 30 degree angle (using my quilting ruler) to use as a guide and I realized that I was working with an isosceles triangle.

After a little googling (alas I don’t remember any high school trig), I found a formula for finding the base of an isosceles triangle when you know the length of the two equal sides and an angle. So exciting! Can you even believe that trig is ACTUALLY USEFUL for pattern making? I can. Also can’t. My mind is a tiny bit blown.

So, if you want to draft with paper using a consistent angle of rotation, please read on. Also, just read on if you’re curious about the math. :)

How to use trigonometry to find the space interval

Here, we are analyzing the triangle of space that is created when we slash and spread the rectangles to make the brim (shown in blue above). We know that the two long legs equal the Brim Radius F. And we get to choose the Angle of Rotation R. With those values we can solve for X the space interval.

To solve for X, we need to divide the triangle in half to create a RIGHT TRIANGLE (shown in blue above). Now, we can apply the SINE function to the right triangle and find the space interval X.

From trig, we know that sin θ = opposite / hypotenuse. Watch this video for an explanation on this formula. For us, the hypotenuse is equal to the Brim Radius F and θ is equal to half the angle of rotation, R/2.

Above, our original rectangle is on the left in gray and the half triangle to the right is in blue. When we input our values into the formula we will use R/2 and X/2 because it is a half triangle. Remember, θ = R/2, the Hypotenuse = F and we are solving for the Opposite = X/2. Our formulas can be simplified as follows:

sin θ = opposite / hypotenuse

sin(R/2) = (X/2)/F

X = 2*F*sin(R/2)

For example, if we use a rotation of R=23° and Brim Radius is F=4.5", the X = 2*4.5*sin(23/2) = 1.79. So, the “slash and spread” can be done digitally using a rotation of 23° or on paper with a spacer of 1.79" and the results should be the same.

Remember, every time you use a different Brim Radius or different Angle of Rotation, you will need to recalculate the Space Interval. If you don’t want to do the math yourself, check out this online calculator.

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I hope you enjoyed this post! I really enjoyed getting into the nitty gritty of the math. This is the kind of post that I never would have thought to share on my sewing blog but it feels just right for this Substack. And it felt like it came about naturally through the process of drafting and testing.

Please let me know in the comments if you have any questions.

Next week, we’ll look at some design variations for our hats. One variation will use arc length and circles and the other will involve some freeform curve drawing.

Happy mathing!

Beth

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