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the point

Ravellings on the knitted sleeve
Part III - Set-in sleeves by the numbers

The last time we looked at sleeves, we looked at why set-in sleeves are shaped the way they are, and how to make a sleeve cap to fit the armscye using either a graph-paper method of plotting and tweaking, or the live seat-of-your-knitted-pants method. Both of those methods give up some amount of certainty in the finished product in order to avoid some computational intensity. However, some designers prefer a more mathematical approach over repeated graphing and measurement, particularly when a set-in sleeve cap needs to be graded (multisized) for a pattern. This article addresses the "trigonometry" comment in the previous sleeve article, complete with sample numbers.

This is shaping of a bell-shaped sleeve cap by calculation and estimation, and may not be for everyone, although the theory is no more difficult than the Pythagorean Theorem. (You remember that, right? If not, there's an appendix with a refresher course.) All that's necessary to accomplish this method is a little time and a calculator with a square root function -- if you're sitting at your computer reading this article, you've got both.

If you really don't want to follow along, jump down to Quicker fixes to completely avoid all the computation, and read a bit about software, short rows, and top-down construction.

The plan

Rules of thumb

You may recall some of these rules from the last column on sleeves:

  • The tighter fitting the sleeve and armscye, the deeper the sleeve cap needs to be. Conversely, the looser fitting the sleeve and armscye, the shallower the sleeve cap will be.

  • The armscye has a vertical depth (from the initial bind-off row to the beginning of the shoulder) and a horizontal depth or indentation (determined by the number of stitches bound off or decreased). The greater the vertical depth with respect to the horizontal indentation, the shallower the sleeve cap will be.

  • The final bind-off row on a knitted set-in sleeve cap usually ranges from 2 to 6 inches. The narrower the sleeve cap, the shorter the final bind-off row will be; it follows, therefore, that with a looser-fitting garment, the sleeve cap will be shallower so the final bind-off row will be wider.

All of the measurements here are presented in inches, without centimetre equivalents -- there are enough numbers in this article without including conversions.

As discussed in the last sleeve article, there are different strategies available to plan the shape of a set-in sleeve cap. You need to begin knowing the maximum width of the sleeve, and the initial bind-off depth at the start of the sleeve cap. After that, you can decide on the sleeve cap height first and create the rest of the cap to fit, or you can decide on some or all of the remaining decrease rates and allow the height to be determined that way. In this procedure, we'll establish the numbers for all the bind-off rows first, then the decrease rates, and worry about sleeve cap height last. The sleeve cap will have a bell curve, and it will likely (but not necessarily) have multiple bind-off rows leading up to the final bind-off row.

We'll proceed in this order:

  1. Set up the sleeve cap with an initial width and known bind-off measurements

  2. Determine the perimeter of the body armscye

  3. Determine a first approximate decrease sequence to fit the armscye

  4. Refine the decrease sequence to create typical set-in sleeve cap curvature

  5. Check measurements, and revise step 4 if necessary

  6. Determine the sleeve cap height and design the remaining part of the sleeve

In order to shape the sleeve cap, we need to know the shape of the armscye first. We'll work with a hypothetical women's pullover with a 40-inch finished bust measurement and a shoulder width of 15 inches, worked at a gauge of 4 stitches and 5 rows per inch and intended to be worn with about 2" of ease. This hypothetical pullover has identically shaped front and back armscyes, which is normal in hand-knitted clothing (it also makes the calculations easier).

In this example, both the front and back armscyes are shaped with the following instructions (these instructions leave out the neck shaping):

Bind off 4 sts at beg next 2 rows.
Dec 1 st ea side for next 2 rows.
Dec 1 st ea side every other row 4 times.
Work even until armhole measures 8 inches.

This decreases the armscye a total of 10 stitches on each side (that's equal to an indentation of 2.5 inches), and provides an armhole depth of 8 inches.

Some knitters may take the 8-inch depth to include the bind-off rows, which means that a further 28 rows should be worked after the bind-off/decrease instructions are completed (28 rows even + 12 rows of decreases and bind-offs = 40 rows, which at a gauge of 5 rows per inch is 8 inches).

Others may choose not to include the two bind-off rows, which means that a further 30 rows should be worked. Provided you're not working at a terribly bulky gauge, two rows won't make much of a practical difference.

In this example, we won't include the bind-off rows, or in other words, "work even for a further 30 rows".


1. Set up the sleeve cap with an initial width and known bind-off measurements

Some numbers need to be arbitrarily fixed. First, the width of the sleeve before the cap shaping must be set. Some general rules of thumb set the ease of a sleeve between 50% and 100% of the ease around the body; in our case, that would be between 1 and 2 inches around the upper arm.

We'll say that the wearer would have a 12-inch upper arm circumference, and set the width of the sleeve at 14 inches and add a one-stitch selvedge on either side, which is equal to 58 stitches at a gauge of 4 stitches to 1 inch. (In this methodology, the selvedge stitch is not treated any differently than the rest of the stitches for the purpose of calculation.)

Next, in this method, we need to fix the final bind-off width. Because we're also planning to have multiple bind-off rows at the top of the sleeve cap, the final bind-off width can be on the narrow side; we'll choose 2.5 inches, which at our gauge will be 10 stitches. This arbitrary value is something that you'll come to set according to your own preferences.

However, if the sleeve is starting with an even number of stitches right before the sleeve cap, then the final bind-off row should consume an even number of stitches in order to be perfectly centred. Similarly, if the sleeve starts with an odd number of stitches, the final bind-off should have an odd number as well.

Finally, the initial bind-off width must be fixed. This will usually be the same as the initial bind-off shaping on the body armscye, which in this case is 4 stitches on each side. In fact, it's almost an unwritten rule for hand knits -- keeping the initial bind-off at the same width as the underarm bind-off makes setting that portion of the sleeve into the armscye easier.

In the course of shaping the rest of the sleeve cap, we might tweak the final bind-off number or the sleeve width by a stitch or two in order to get all your numbers to fit. However, the initial bind-off width will never be changed unless we decide to change the bind-off width on the body.

At this point, we can deduce that, given these bind-off rows, a further 20 stitches must be decreased or bound off on either side of the sleeve cap.

Now it's time to calculate the perimeter of the armscye, which is where the trigonometry comes in.

2. Determine the perimeter of the body armscye

The armscye can notionally be divided into different shaping zones: one for each region having a different bind-off or decrease frequency. We need to calculate the approximate length of the garment edge for each of these zones. In our hypothetical sweater, each side has four such zones:

Zone Decreased/
bound-off stitches
Number of rows
1 4 stitches 0 rows (these are the bind-off rows, and we'll ignore their height for the purpose of calculation)
2 2 2 ("dec 1 st ea side for next two rows")
3 4 8 ("dec 1 st ea side every other row 4 times")
4 0 30 ("work even until armhole measures 8 inches")
total 10 40 rows

First we convert these stitch and row numbers to inches, based on a gauge of 4 stitches and 5 rows per inch:

Zone Decrease/
bind-off length
Height (depth)
1 1 inch 0 inches
2 0.5 0.4
3 1 1.6
4 0 6
total 2.5 inches 8 inches

Now, we need to calculate the perimeter associated with each zone (refer to the appendix on the Pythagorean Theorem if necessary):

Zone Decrease/bind-off length
1 1 inch
2 0.64
3 1.89
4 6
total 9.53 inches

Thus, half an armscye -- either the front or the back half -- has a perimeter of 9.53 inches. The total perimeter of the entire armscye, then, is 19.06 inches. This means that the perimeter of our sleeve cap, if it is to fit the armscye, must measure 19.06 inches as well; it may even be longer than that, say about another inch longer, and it will still fit into the armscye without noticeable puckering. Therefore, we'll aim for a sleeve cap with a perimeter that measures between about 19.06 and 20.06 inches.

Because our armscye and sleeve cap are symmetric, we can work in terms of a half-armscye and half-sleeve cap. Therefore, we want to create a sleeve cap with a half-perimeter measuring between 9.53 and 10.53 inches. We certainly won't be designing a sleeve cap to this level of precision; an approximate number like 9.5 or 10.6 inches will be fine.

You might have figured out that the real lengths of zones 1 and 4 are not exactly 1 inch and 6 inches, but could be more or less depending on how the side and shoulder seams are joined. The side seam will not be an issue, because the initial bind-off in zone 1 will be matched by the sleeve cap shaping, and the sleeve seam will have the same seam allowance as the side seam. As for the shoulder seam, that's not enough to make a significant practical difference.

3. Determine a first approximate decrease sequence to fit the armscye

Now, to fill in the big blanks in the sleeve cap shaping. In this method, we'll follow some rules:

  • No more than one decrease per edge in any row. For example, "dec 1 st ea side for next 4 rows" is acceptable, but "dec 2 sts ea side for next 2 rows" is not. If your computations suggest that you need to work more pairs of decreases than you have available rows, then you will need to either expand or increase the number of bind off rows, by adding another pair of bind-off rows at the bottom or top of the sleeve cap, or by increasing the number of stitches bound off at the top of the sleeve cap. The only bind-off row that is inviolate is the initial pair of bind-off rows: the ones that match the initial pair of bind-off rows on the body.

  • It's preferred, but not mandatory, to keep easy-to-follow decrease rates on the sleeve cap, such as every row, every other row or every fourth row. This keeps as many decreases as possible on the right side of the fabric, because this seems to make instructions easier to follow for some knitters. If it's simply not possible to follow that rule, then go ahead and decrease on every third row. ("Every third row" is easier for a knitter to follow than "every second and fourth row alternately", although "every second and fourth row alternately" does keep all decreases on the right side.)

  • If you find that you need to do a lot of tweaking to your sleeve cap, and you really need guidance on where to end one set of decreases and start decreasing with a different frequency, try following the general rule of keeping the point of inflection on the sleeve cap shape in approximately the right place.

Point of inflection?

The typical set-in sleeve cap gains its bell-like shape from a change in the curvature from positive to negative. This change is an inflection point. For those of you who love calculus, no further explanation of an inflection point is needed. For those of you who don't, it's the point where the sleeve cap stops being concave upwards (around the underarm) and starts being concave downwards (around the shoulder). For those of you who didn't like the word "concave", the point of inflection is the big blue dot in the diagram. See?

That point corresponds to the point on the armscye where the decreasing stops and the straight vertical part starts. In practice, the curvature of a hand-knitted sleeve cap doesn't always immediately switch to an obvious negative curvature at the point of inflection; it just stops being obviously positive by that point. Furthermore, the bulkier your gauge, the less scope there is for refining the curvature of the sleeve cap, because you simply don't have enough stitches and rows to follow a precise curve. (Think of intarsia--if you tried to knit a graphic design in a fixed area, you get a lot more detail if you knit at a finer gauge.) So, when drafting the sleeve cap, if we take the point of inflection into consideration, we won't aim for absolute mathematical certainty; we'll just aim to keep the point of inflection in that neighborhood.

In our hypothetical sweater, the point of inflection occurs at the junction of zone 3 and zone 4. So, a half sleeve cap to fit this half armscye will curve around the underarm, but that curvature will have stopped by the time the perimeter equals 3 inches.

Now, how would these underarm and shoulder curvatures be represented in knitting? If you described the underarm curvature in words, you'd say that it starts out flat, begins a shallow curve, and gradually climbs so that the angle gets steeper and steeper. The shoulder curvature starts out steep, but gradually eases off to be shallow, then becomes totally flat at the final bind-off. If you described it in numbers, in our bell-curve-like sleeve cap you'd say that the tangent of the sleeve cap curve starts at zero degrees and increases to a maximum angle, then decreases again to zero. Either way, in the context of knitting a sleeve cap:

  • A flat line (zero degree tangent) corresponds to a bind-off.

  • A very shallow curve corresponds to multiple decreases in each row(preferably, a bind-off row or two).

  • A shallow curve corresponds to one decrease in every row.

  • As you reduce the frequency of decreases to one decrease every other row, to every three rows, four rows, etc., the curve becomes progressively less shallow, or progressively steeper.

Filling in the numbers: first attempt

First, let's add a little more shaping to the top of the sleeve cap. We've already fixed a final and an initial bind-off width. If you were to picture the sleeve cap in your mind's eye at this point with a constant decrease rate between the bottom and top of the sleeve cap, you could imagine that there'd be a rather sharp corner at the very top -- not very curvy. This is one area where we'd like a very shallow curve, so we could add some further, shorter bind-off rows leading up to the final row to soften the angle.

A good first attempt for this area is to add 50% of the final bind-off width to either side of the final bind-off row, and to split this 50% over a couple of bind-off steps -- maybe evenly, or maybe in a roughly two-thirds/one-third proportion. In our hypothetical sleeve cap, we had a final bind-off of 10 stitches (2.5 inches), so we'll add some more bind-off rows that remove a further 5 stitches (50%) on either side. And we'll split that up over a couple of steps, which in this case is 3 stitches + 2 stitches. If you graphed this sleeve cap, it would look like this [see below]: (if you were using this method of calculating sleeve cap shaping, you likely wouldn't bother plotting it out; these graphs are provided for illustration)

The corresponding sleeve cap instructions would read like this:

[instructions from wrist to sleeve cap shaping]
58 sts on needle.
Bind off 4 sts at beg of next two rows.
[decrease instructions to be filled in]
Bind off 2 sts at beg of next two rows, then 3 sts at beg of foll two rows.
Bind off remaining 10 sts.

We're now left with 15 stitches to decrease on either side of the sleeve cap. We need to figure out how to do that, while keeping the half-perimeter of the sleeve cap between 9.53 and 10.53 inches.

Each half sleeve cap has an initial bind-off of 4 stitches (equal to 1 inch of the perimeter) and a final bind-off of 5 stitches at the very top (equal to 1.25 inches). We've now added further bind-offs of 5 stitches over 4 rows. In terms of inches, that's a length of 1.25 inches over a height of 0.8 inches. The approximate perimeter measurement of this segment is about 1.48 inches. Therefore, in total, the bind-off rows in the above diagram take up about 3.73 inches of a half-sleeve cap perimeter (1 + 1.25 + 1.48 inches). This means that the remaining part of the half-sleeve cap, in which the 15 stitches must be decreased, must cover a length of between 5.8 inches (9.53 - 3.73 inches) and 6.3 inches.

For a first estimate, we'll calculate what average decrease rate will give us the perimeter we need. 15 stitches to be decreased is equal to 3.75 inches of width. If this was one short leg of a right triangle, and the 5.8 inches was the hypotenuse, then the height of that triangle would be approximately 4.42 inches. In our gauge, that's 22.1 rows (we'll make use of some of the half inch of tolerance we have in determining the perimeter of the sleeve cap, and round that up to 23 rows). In other words, then, if we decreased 15 stitches more or less consistently over 23 rows, we'd fit our sleeve cap.

To figure out the decrease frequency, we'll divide 23 rows by 15 stitches to arrive at 1.53 rows per stitch: in other words, if we worked a decrease approximately every 1.5 rows, we'd fit our target perimeter measurement. In this case, decreasing one stitch every 1.5 rows is the same as decreasing 2 stitches every three rows, or something like "decrease 1 st every other row and every row alternately". If we used this decrease, the instructions would look like this:

[instructions from wrist to sleeve cap shaping]
58 sts on needle.
Bind off 4 sts at beg of next two rows.
Decrease 1 st ea side every other row and every row alternately until 20 sts remain.
Bind off 2 sts at beg of next two rows, then 3 sts at beg of foll two rows.
Bind off remaining 10 sts.

Now, we could stop there, and use this type of decrease for our sleeve cap. We'd wind up with a relatively simple trapezoidal shape, and if plotted on proportional graph paper, it would look like this:

That's a roughly 5.4 inch high sleeve cap. And it doesn't look too bad for a first attempt.

4. Refine the decrease sequence to create typical set-in sleeve cap curvature

Even though the first attempt at a sleeve cap looks pretty good, it's not the best. We could leave the sleeve cap as is and move on to the next step, but from a pattern-writing perspective, working decreases "every other row and every row alternately" is annoying to write and frustrating to follow. Also, depending on fitting preferences, this sleeve cap might look too shallow or seem a little too trapezoidal.

If the decrease rate was the only concern, we could address it by splitting up the every-row decreases and every-other-row decreases, so that every-row decreases are worked first, and the decreases every other row second. Currently, our "decrease 1 st ea side every other row and every row alternately until 20 sts remain" instructions result in 8 every-other-row decreases, and 7 every-row decreases. If all the every-row decreases were worked first, the sleeve cap edge would look like the outline on the left, below. But we could also work all the every-other-row decreases first, as shown in the middle outline. We could even split up the decreases, and work some every-row decreases, then the every-other-row decreases, and then the rest of the every-row decreases, as shown below.

So how would we divide up these decreases? If our objective is a shape resembling a bell curve, we can see by comparing the outlines above that the rightmost shape is the most desirable; from the initial bind-off, the sleeve cap begins with a shallow curve, gets steeper, then tapers off with another shallow curve before the final bind-off row. If we used this shape, our instructions would read:

[instructions from wrist to sleeve cap shaping]
58 sts on needle.
Bind off 4 sts at beg of next two rows.
Decrease 1 st ea side every row 4 times, then every other row 8 times, then every row 3 times.
Bind off 2 sts at beg of next two rows, then 3 sts at beg of foll two rows.
Bind off remaining 10 sts.

We could take that and move on to the next step. But if you're more obsessive than this, and you really wanted to read the words "point of inflection" again, we could try to place that transition from every-row to every-other-row decreases at the approximate point of inflection.

To figure out how to make the decrease frequency change at the point of inflection, first we need to figure out what portion of the perimeter leads from the underarm to the inflection point. As we figured out earlier, that portion of the perimeter is 3 inches; the first inch of that consists of a bind-off edge, both on the sleeve cap and the armscye. This means that the first set of "decrease 1 st ea side every row" must be carried out over enough rows to equal a perimeter of 2 inches. To do this, we'd figure out the approximate length of a single decrease over a single row (that's the diagonal of 1 stitch and 1 row, which at our gauge is equal to about 0.32 inches), and figure out how many of those lengths go into 2 inches (the answer is approximately 6.25; let's round that down to 6). This means that 6 of the every-row decreases would be worked first, then the set of every-other-row decreases, and then the last every-row decrease would be worked again:

[instructions from wrist to sleeve cap shaping]
58 sts on needle.
Bind off 4 sts at beg of next two rows.
Decrease 1 st ea side every row 6 times, then every other row 8 times, then every row once.
Bind off 2 sts at beg of next two rows, then 3 sts at beg of foll two rows.
Bind off remaining 10 sts.

We've eliminated the annoying decrease instructions, and we've given the sleeve cap a more bell-like curve. If there are other things to be tinkered with, such as the sleeve cap height, this is the time to do it. Remember that we can add up to about half an inch to each side of the sleeve cap without seriously affecting the assembly of the sweater, so an easy way to sneak in a bit more height is to simply add a couple of rows. We'll do it in this example, just for fun. We'll turn two of those every-row decreases into every-other-row decreases. This will shift the point of inflection on the sleeve cap downwards. That's okay -- we didn't move it too far.

And while we're at it, see those instructions to bind off 2 stitches at the beginning of two rows? That's just silly, because while it's equivalent to decreasing 1 stitch at each end for two rows, it has the disadvantage of adding a stair-step edge to the sleeve seam. We put those bind-offs in when we worked on the final sleeve cap shaping; they're not mandatory. This is a good time to turn them into decrease instructions.

Now, our instructions read like this:

[instructions from wrist to sleeve cap shaping]
58 sts on needle.
Bind off 4 sts at beg of next two rows.
Decrease 1 st ea side every row 4 times, then every other row 10 times, then every row 3 times.
Bind off 3 sts at beg of foll two rows.
Bind off remaining 10 sts.

And our sleeve cap looks like this:

5. Check measurements, and revise step 4 if necessary

It's not necessary to check your measurements over again if you simply used that first approximation of the sleeve cap. In this example, it probably isn't even necessary to double-check that the sleeve cap perimeter is no more than an inch longer than the armscye perimeter--all we did was rearrange some decreases, then stick in a couple of more rows. However, if you do any further tweaking, and if you decide to alter the width of bind-off rows, or adjust the height of the armscye or the width of the sleeve before the sleeve cap shaping, it's a good idea to make sure that the sleeve cap will still fit the armscye. You may find that you need to undo some of your tweaks.

6. Determine the sleeve cap height and design the remaining part of the sleeve

We're just about done. All that remains is to figure out how to knit the rest of the sleeve, which is a lot easier than what we just did. For this, we need to know the total length of the sleeve, and the width at the bottom. This is covered in the first and second sleeve articles.

And that's it.

Quicker fixes: software, short rows, and knitting from the top down

You can avoid the entire process of calculating armscye and sleeve cap shaping on your own by using software designed to generate knitting patterns based on your input measurements. It's up to you whether the savings in labour is justified by the expense.

Depending on your design needs, some knitting software may not give you the degree of control that you want over the precise fit and silhouette of a set-in sleeve. You'd need to explore each program's capabilities to see if you can adjust the sleeve cap height or final bind-off width, for example. If possible, try a demo version of the software first, to see if it will generate the type of set-in sleeve that you want.

Next, a word on being clever with short row shaping on a sleeve cap: you may be familiar with short-rowed shoulders, which make seaming slanted shoulders a snap by eliminating the stair-step effect caused by repeated bind-off rows. It might occur to you to do the same thing with a sleeve cap worked from the bottom up, since a smooth edge is easier to work with than a jagged edge, and in fact, it's a good idea.

However, you need to exercise caution: working short rows creates an angled line of live stitches (a hypotenuse, if you will), longer than the width of the same number of stitches worked straight in an unangled row. Finishing a series of short rows normally involves knitting across all stitches, hiding any wraps, using the same number of live stitches. This will cause the short-rowed area to draw in, or cup (such as in short row bust shaping). If this happens at the top of a sleeve cap, the cap will curve over the shoulder like a tailored suit -- but the final sleeve cap perimeter will be too tight to fit the armscye, unless you increase a few stitches when knitting the final full row to flatten the sleeve cap out to its originally intended dimensions.

If you want to deliberately curve the sleeve cap in this manner, then the sleeve cap will need to be resized to fit the armscye, and you should first ensure that the shoulder seam will hit the shoulder at precisely the right point. Otherwise, the sleeve cap will cup too low on the arm or too high on the shoulder, like a misplaced puffed sleeve.

And speaking of short rows, a short-row method is used by Barbara Walker in her book, Knitting from the Top, to create top-down set-in sleeves that are picked up from the armscye and worked down towards the wrist. In essence, the stitches required to meet the maximum sleeve width are picked up around the armscye; the sleeve cap is shaped by beginning with a fixed proportion of stitches centered at the shoulder, and working short rows that consume one additional stitch at the end of each row until the bottom of the armscye is reached. In bottom-up terms, that's equivalent to a decrease of one stitch at each end on every other row between the initial bind-off rows and a single, final bind-off row. This creates a simple trapezoidal sleeve cap rather than a more complex bell shape.

Like the computationally intense method we worked through above, Walker's method doesn't bother establishing the height of the sleeve cap before figuring out the shaping; it sets the width at the top of the sleeve cap, and the width of the lower horizontal portions of the sleeve cap outline, then fills in the rest. However, unlike the method used in this article, Walker's method is not concerned with working a sleeve cap that matches the perimeter measurement of the armscye; rather, the sleeve cap is "forced" to match the perimeter measurement, because the sleeve cap stitches are picked up around the armscye using whatever stitch-to-row proportion is necessary to distribute the stitches evenly among the rows of the armscye. The sides of the sleeve cap trapezoid fit neatly into the curved armscye portions because the stitch gauge of the sleeve cap is distorted when the stitches are picked up around the armscye.

The Walker top-down sleeve has a relatively wide upper edge (the edge that would normally be the final bind-off edge when knitting from the bottom). In the proportions used by Walker, the upper edge of the sleeve cap is one-third of the maximum sleeve width. Thus, for the sleeve width in our hypothetical sweater example, Walker's upper edge would measure about 4.7 inches straight across, whereas we had 2.5 inches (softened with a further pair of bind-off rows). Strictly following Walker's methodology also yields a strange result: you'd discover that the narrower the sleeve, the shorter the sleeve cap will be, and the wider the sleeve, the longer the sleeve cap will be. This contradicts the general rule of thumb that a close-fitting sleeve should have a deeper sleeve cap. Theoretically, the effect of strictly following the Walker method in a tight, short sleeve would be a sleeve hem that does not fall horizontally across the upper arm; instead, the sleeve fabric would be stretched by the bulk of the shoulder and upper arm, and the sleeve hem would appear curved or arched over the upper arm. (This might produce the impression that your upper arms are either muscular or fat. But that depends on your own perspective.) Such fitting concerns could be addressed by altering the proportions of the stitches picked up around the armscye, and changing the lengths of the short rowsto produce a more rounded cap.


Walker, Barbara, Knitting from the Top [Schoolhouse Press, 1996].

And see the references in the previous two Thinking Beyond the Pattern columns on sleeves [1, 2].

Appendix: Pythagoras says...

In a right-angled triangle, the square of the length of the hypotenuse (H) is equal to the sum of the squares of the other two sides (S and R). In the diagram to the right, you can see that the squares of the two shorter sides is equal to the square of the longest side (9 + 16 = 25).

So, if you've got your trusty calculator, here's how you'd figure out the following numbers:

1. If you know S and R, figure out H like this:

  • calculate the square of each of S and R

  • add the square of S to the square of R

  • take the square root of that result to get H

Most calculators will give you the answer if you press these buttons: [where you see "S" or "R", type in the number that you have for those values]

On your calculator, the square and square root functions may be on the same button, and one will require you to press a "shift" or "inv" key first.

2. If you know H and S, figure out R like this:

  • calculate the square of each of H and S

  • subtract the square of S from the square of H

  • take the square root of that result to get R

And most calculators will give you the answer if you do this:

3. If you know H and R, figure out S by using the same logic. The calculator instructions would run like this:

The Pythagorean Theorem applies handily to knitting, because most knitting creates a grid of stitches, in which the rows are perpendicular to the columns. The two shorter sides of a knitted right-angled triangle can be represented by the number of stitches and rows; the hypotenuse, which is the longest side, runs diagonally, and could roughly follow an increase or decrease edge.

A slight confounding factor arises in that the height of a stitch usually isn't the same as the width of the stitch, but that can be dealt with by converting all stitches and rows to inch (or centimeter) measurements.


Public speakers know that you lose half your audience with every equation you show.

Jenna can't believe you read this far.

Find more cures for insomnia at her website, the girl from auntie.