Here you have a curvy quadrilateral bounded by arcs of two concentric circles and two radii of the circles. If you cut this shape out and sew the straight edges together, you will get a slice off the bottom of a cone.1
Since this shape is used to make hat crowns, hat brims, skirts, flounces, collars, cuffs, dunce caps, and ruffles, it is worth a little time to learn how to choose what angle to put between the two straight edges, and what radius to use for each circle.
If this isn't a Barbie skirt, you will almost certainly want to draft a pattern for half a circle.2 If you are making a half-circle skirt, this will be the entire pattern, if you are making a full-circle skirt, it will be half of it, if you are making a circle-and-a-half, it will be a third of the pattern, and so on.
So the inner arc will be your waist measurement, half the waist measurement, a third of the waist measurement, or a quarter of the waist measurement. It will also be half the circumference of the circle that you want to draw. What you need to know in order to draw this circle is the radius.
Conveniently, the whole circumference is twice the radius times pi, which means that the half-circle arc is pi times the radius. So to get the radius, divide the length of the arc by pi. If there isn't any pi on your calculator, use 3.14. If you are dividing by hand, use 3 1/7 or 22/7.
To get the radius of the outer arc, add the desired length of the skirt to the radius of the inner arc.
The odds are that you haven't got a compass big enough to draw a circle skirt. The traditional solution to this problem is to tie a string to a pin, tie a loop at the other end, and put the point of a pencil into the loop.
Another way is to use a yardstick to mark a great many points along the arc, then connect the marks freehand. If it's a long skirt, use a carpenter's folding ruler.
When you add seam and hem allowances, the cutting lines of the waist and hem will be circles concentric with the seamline and the hemline, but the cutting lines parallel to the radii will not themselves be radii -- unless you don't mind wedge-shaped seam allowances.
A flounce is a wide ruffle sewn to the bottom of a skirt, but you can also think of this style as a circle skirt with a very wide yoke, and use the hem-length of the straight skirt as the waist measurement in the instructions for a circle skirt.
You can also use a straight-skirt pattern to design a circle skirt — or any other style of skirt — with a yoke.
But the angle is unlikely to be a nice neat half circle. Let us suppose that you measure a hat you like, and determine that a circle with a fifty-degree wedge snipped out of it will give you the desired slant.
Snipping fifty degrees out of three hundred and sixty would leave 310O. 310/360 times the circumference of the desired circle is equal to your head measurement.
So to get the radius, you would multiply your head measurement by 360 and divide the result by 310 to get the whole circumference. Dividing the circumference by pi gives you the diameter, and dividing the diameter by two gives you the radius.
If you are calculating by hand, you would first simplify 310/360 to 31/36, so that you could multiply by 36 and divide by 31. Then you would note that you are multiplying by an even number and later on will divide the result by two, so you'd save a step by multiplying by 18, after which you'd divide by 31, then divide by pi.
Of course, if *I* were calculating by hand, I'd decide that a forty-five degree snip was *quite* enough -- I know without any calculation that taking 45O out of 360O leaves 7/8 of a circle. Then 8/7 of my head measurement would be the circumference of the circle, and 4/7 of my head measurement divided by pi would be the radius.
If you want to make a fez-shaped hat crown, the outer arc is your head measurement. The height3 of the hat is the difference between the radius of the outer arc and the radius of the inner arc. The inner arc is the circumference of the circle sewn in at the top of the hat.
One way to approach this problem is to roll a piece of paper to the desired slant, tape it, cut it open, and measure the angle between the cut edges. Then you can make the same calculations for the outer arc of the hat crown that you made for the inner arc of the hat brim, draw the pattern, and measure or calculate the length of the inner arc to find out how big to make the circle you sew in to make the top of the hat.
Or it may be that you know how big a circle you want for the crown. In this case, you will have to resort to algebra.
You begin by calculating the circumference of the crown circle. This is the length of the inner arc. You now know the length of the outer arc, the length of the inner arc, and the distance between them.
You also know that the outer arc is to its radius as the inner arc is to its radius, and that the distance between them is equal to the outer radius minus the inner radius.
Or, to put it another way, the outer radius is equal to the inner radius plus the height of the hat.
In the interest of keeping statements simple, let us
call the height of the hat D
(for "difference" or "distance")
call the length of the inner arc L
call the length of the outer arc M
(for "I've already used 'L', so I took the next letter.")
and call the radius of the inner arc R.
We know that
L/R = M/(R+D)
and we need to know R. Whapping the equation around:
L = MR/(R+D)
L(R+D) = MR
LR + LD = MR
LD = MR - LR
LD = R(M-L)
LD/(M-L) = R
So if you multiply the length of the inner arc by the distance between the two arcs, then divide the result by the difference in length between the two arcs, you get the radius of the inner arc. (Make sure everything is measured in the same units!)
You can then calculate the angle by finding out what fraction of its circle either arc is. Both arcs should give you the same fraction, which provides you with a check on your work.
If you want the entire cone, things are considerably simplified. The length and radius of the inner arc are both zero, so the difference between the inner radius and the outer radius is the outer radius.
What you are left with is the outer arc, its radius, and an angle. Knowing any two of these three will determine the figure.
Since knowing two out of three is the same as leaving out one of three, it is obvious that there are only three possibilities for the information you start with.
Arc length and angle -- most likely when you want to make a dunce cap, wizard hat, witch hat, or hennin. The outer arc is the head measurement, plus whatever ease is desired. You get the angle by rolling a paper cone of the desired shape, opening it flat, and laying a protractor on it. The hat-brim discussion tells you how to get the radius from the angle and arc length.
Arc length and radius. Here you calculate the circumference of a circle, calculate what fraction of that circumference your arc is, and that's your angle.
Radius and angle. No problem, just draw it.
When you make a long ruffle, your angle will be many full revolutions, so you will make a bunch of circles or half-circles and sew them together.
Let us first discuss the simpler case in which you piece a ruffle longer than the thing to be trimmed, then trim the last piece to make it fit.4 Working ad hoc this way is apt to cause the ruffle to be eased on, because the cut edge is shorter than the seam line, but as long as you don't sew in any pleats or puckers, extra fullness in a ruffle is all to the good. When a ruffle is really long, you can calculate the number of pieces needed, round that up to a whole number, and then make it fit.
Whether you are drawing a half circle or a full circle, all you need to know is the radius of each arc.
If you want the outer edge to be twice as long as the inner edge, the outer radius is going to be twice as long as the inner radius. Draw the inner arc with a radius equal to the desired width of the ruffle, then draw the outer arc with a radius twice that. Add seam and hem allowance.
If you want the outer edge to be three times as long as the inner edge, draw the inner arc with a radius equal to half the desired width of the ruffle, then draw the outer arc with a radius three times that.
If you want a fullness of two and a half
-- urk, back to Algebra.
(If you don't speak algebra, skip down to the line that begins "So")
Use "W" to represent the width of the ruffle,
"F" (fullness) for the ratio of the outer arc to the inner arc,
"r" for the radius of the inner arc.
and "R" for the radius of the outer arc.
Now the outer radius is the fullness times the inner radius: R = Fr
and the outer radius is the inner radius plus the width of the ruffle: R = W + r
So if you know the inner radius, the outer radius is nailed down twice.
Since two things equal to the same thing are equal to each other,
Fr = W + r
Fr - r = W
r(F-1) = W
r = W/(F-1)
So to get the inner radius, you subtract one from the fullness and divide the width of the ruffle by the difference.
for a fullness of 2/1, r = W/1 -- and in our first example, the inner radius was equal to the width.
for a fullness of 3/1, r = W/2 -- and in our second example, the inner radius was half the width.
for a fullness of 2.5/1, r = W/1.5 -- so the fullness we set out to find would require an inner radius of two thirds of the width of the ruffle.
If calculating by hand, it is easier to say five to two instead of
two-and-a-half to one.
One is equal to 2/2, so 5/2 - 2/2 = 3/2
and as you recall from third grade, you divide by a fraction by simply flipping it over, so W divided by 3/2 is 2/3 of the width.
Unless the fabric is very wide, or the skirt is very small,
a circle skirt is nearly always made by sewing half circles together.
If you use an angle which is not a multiple of right angles, some of the seams are going to slant across the grain, which is an un-necessary complication. Half circles are as economical to cut as quarter circles -- more economical, since you save two seam allowances -- and with quarter circles, you'd have the additional trouble of keeping track of which straight edges are on the straight grain and which are on the cross grain.
One problem with circle skirts is that they hang on the straight grain, the cross grain, the bias, and every angle between. This makes them somewhere between difficult and impossible to hem evenly. This may be why felt was a popular fabric for the poodle skirts of the fifties!
To solve this problem, and also avoid the large amount of waste involved in cutting circle skirts, one could cut a piece of fabric into a large number of trapezoids, with narrow ends that add up to the waist measurement, turn them so that the narrow ends all point the same way, sew them together, then trim off the corners. The section describing this process has not yet been written. (And won't be written until I find an excuse to make myself a trapezoid skirt.)
 This is the height measured along the slanting side of the crown, of course. If the angle is large, there will be a considerable difference between the height measured along a side and the height measured straight up. For a fez, there is not much difference. If you are making a coolie hat, what I've called "height" is more like "width" -- as in hat brims.
 It's better to trim the same amount off both the first and the last piece to put both cuts on the same grain, which makes it easier to sew the seam neatly. On the other hand, this is a very short seam -- and if the grain isn't the same on both sides, who's going to notice?
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