Triangles and Triangular Shawls

I thought I’d post on some of the maths I use when knitting or crocheting triangular shawls. It starts with something called similar triangles. If that scares you just remember that triangles can be nice – like this cute little one:

Two triangles can be similar as well as nice. What does that mean? It means they are the same shape, but not the same size. If you start with two identical triangles and shrink or magnify a triangle (keeping the proportions the same) then the two triangles are similar. The two triangles have the same angles and the smaller one sits perfectly inside the larger:

Now the area of a triangle is half the product of the lengths of two sides, multiplied by the sine of the angle between them.

Suppose the ratio of two similar triangles is 2:3, meaning the length of each side of the smaller triangle is 2/3 the length of the larger triangle. If the large triangle has the area in the formula above, then the area of the smaller triangle will be

So that the small triangle has 4/9 the area of the larger one.
But how does this apply to knitting? Imagine you’re knitting a triangle in rows from the point down to the base. At some stage you have knit as far as the line in the picture above. So the triangle you have knit is similar to the whole triangle that you plan to knit. The area of the triangle is a good estimate for the ‘amount of knitting done’ and also the metres used. But all knitters want to know is the portion of the entire project they have done – something that follows easily using similar triangles:

Most triangular shawls can be thought of as two triangles which are joined down the ‘backbone’ or centre of the shawl. Like in this next picture where the lines are the current row of knitting and the shawl increases in the direction of the arrows.

Since we multiply both the smaller area and the larger by 2, the relative difference doesn’t change and the same formula works! So at any stage in a shawl you can calculate how many rows you have knit versus how many are in the pattern, multiply that fraction by itself and you know how much of the shawl is knit!

This is useful for making the most of every metre of yarn – especially when that yarn is very precious or you want to make a shawl as large as possible. Simply weigh the ball after you wind it but before you start knitting and make a note of it, say on the ravelry project page. Weighing the actual ball you will be working with is important as not every skein is exactly the weight on the tag. I also use the yarn tag as a core for the ball which adds extra weight, if that’s the case it’s good to weigh the tag too and take note so that it isn’t mistaken for remaining yarn later! Then start knitting. At any point, say before deciding to knit an extra repeat of a chart, simply count the rows completed and the total rows in the pattern with the repeat added. Then calculate what fraction of the shawl is knit. Weigh your yarn ball next and provided you’ve used less than that fraction of yarn then you can knit the extra repeat. Or you could use this handy formula to see how many more rows you can knit with the yarn you have left. Provided the ‘rows left’ is more than the rows remaining in the pattern then all is fine.

All of this comes with the MAJOR CAVEAT that these formulae do not account for nupps, bind offs or anything else that takes extra yarn. They are only for estimating whether you’ll have enough to finish with one ball, whether to add extra repeats and how many to do.

There are no specifications on the triangles in the shawl so these methods can be used for the Ishbel shawl, even though that pattern increases twice as quickly at the edges than at the centre. Any shawl that can be knit by increasing along the same lines can be described by triangles. For semicircular shawls a similar area-proportion calculation can be done too. I might share that another time!

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