Let’s talk hexagons or, as the fabric people like to call them, hexies! They look cute and they make up a solid structure used by nature, but oh the mathematics. Since I have yet to use these beasts in my own sewing, I haven’t really grasped which part of the hexie is the 1” or 3” or 1/2” even. Until now, that is :) While you might consider not reading more, please do, because you will be rewarded. It isn’t horribly bad at all!
Some useful hexie geometry
So, what is a hexagon? The name stems in Greek where hex- or hexa refers to six and γωνία, gonía, means corner or angle. Yup, six corners. Since some kind soul has made a bunch of fantastic pictures available on Wikipedia in the public domain, I’ll use some of them here. Click on the picture to go to the article in question, if you want more information, as it’s quite fascinating once you get over the possible initial primal-scream phase.
There are six sides to a hexagon, which is expressed like n=6 (think of n as number). Since a hexagon is a “regular polygon”, all its six sides are of equal length. If you look closely, you’ll see that the circumradius (the radius) of the circumscribed circle (grey circle) is that long, too.
In other words, the side, s, of the hexagon equals the circumradius, R: s = R. A 2-inch hexie template has a 2-inch side and also its radius is that long.
Since all sides are equally long, the internal 120° angle is the same at all six vertices (sing. vertix), corners or intersections, of the hexie. This in turn means that two radii starting from vertices closest to one another will create an angle of 60° when they meet in the circumcentre, the centre of the circle and, hence, hexagon.
The parts mentioned above work for other polygons, too, such as this pentagon (I’ll get to “a” below):
From the pentagon picture you can see that the diagonal of a hexagon is simple to calculate. You only have to double the length of the circumradius (d = 2R), because it is drawn from a vertix through the circumcentre and out to the opposite vertix. Some quilters like to work with pentagons, though, and while the radius is like drawn in green above, there are actually five diagonals, which create a star pattern when you draw from any given vertix to its two opposing vertices. Try drawing if you like, it’s fun. None of the diagonals cross the circumcentre in that case and I hear a distant voice calling foundation paper piecing… But back to the hexagon!
This means that when you measure with a ruler the side of a hexie template, you’ll instantly know the length of its diagonal, because d = 2R = 2s. Recall that the side of the hexie is as long as its radius. The diagonal can be useful in quilting, which I’ll talk more about below.
The apothem, a, is the distance from a circumcentre to a side when they create a 90° angle (are perpendicular). It is even more important to understand than the diagonal. See for yourself how it looks in a hexagon:
If it helps, imagine this hexie rotated once, so that the green line is actually pointing at 12 o’clock. A quilter might instantly keep drawing the line down to 6 o’clock and perhaps even add a seam allowance or two :) I’ll talk more about the practical implications in a bit.
So how do you figure out the length of the apothem? By using this formula (or by relying on friendly people, who have assembled a table for you):
Recall that a stands for apothem, s is the side and R the radius. You also know that n is 6 for a hexagon (5 for a pentagon). The trigonometric functions cosinus and tangens (cosine and tangent if you’re English-speaking) are to be found on your calculator, as is the number pi.
In quilting, it’s much easier to measure the side of a random template and proceed to use the first portion of the equation. (The alternative is to find the middle of the hexagon through drawing all diagonals and then measuring the radius.) Once you have figured out your apothem and doubled it, you have the height of the hexie when sitting on one of its sides.
If you’re interested in doing the process backwards, you can do that too. Define how tall you want your hexie to be and call that number 2a. Then divide it by two to get the apothem. The side of the hexie you will need will be: s = a × (2 tan(π/n)) = a × (2 tan(π/6)). Again, this would be convenient to read from a table, but learning how to use the formula will open all the doors.
As a conclusion, regarding hexie anatomy, there’s the side, diagonal and apothem to understand. Hexie “relationships” on the other hand are the angles. For quilters the world of polygons is highly interesting, because there is so much regularity, at least when it comes to the regular polygons of both concave and convex kind. I see lots of opportunities in both English and foundation paper piecing to tackle challenging angles. Just look at this concave hexagon:
Hexies in quilting
All of this is so very useful if you start by cutting width-of-fabric (WOF) strips of a particular height and then proceed to cut with the help of a template the individual patches for hexagons or pentagons. Here’s a fabulous tutorial by Pretty by Hand where she shows how to cut huge amounts of hexies in a quick and efficient manner using WOF strips.
You can place the hexie sitting on its side, with sides parallel to the edges of the WOF strip, and with the vertices of two hexies touching, like in the Pretty by Hand tutorial and here:
The height of a piece of fabric needs to be two apothems plus two seam allowances.
Or you can place the hexie on a tip, a vertix, so that the seam allowances of the sides of two hexies will touch. In that case the height of the strip is the hexie diagonal plus your seam allowance of choice, probably slightly less than a quarter inch. Recall that the seam allowance calculation primarily concerns the sides of the hexie, as the corners (vertices) will be wrapped well into two layers once you fold around both sides.
This method will require more cutting than when the hexies sit on the side, but if you have a wider strip, which you want to use to full capacity, you can crank more hexies out of it this way. By this I mean that if you place a hexie on its tip rather than side, you can make smaller but more hexies, when using the same strip height. It can be a bit confusing, but draw two lines like were they part of a WOF strip, then draw a few hexies sitting on vertices, and finally a few more sitting on their sides. See?
Let’s look at an example to make this clear. If you wish to sew with 2-inch hexies sitting on the vertix, remember the side is 2”. This translates to the diagonal being 4”. If a strip of fabric is even slightly larger than 4”, less than 4 1/2”, the hexie template should fit beautifully on it with room to fold the fabric nicely.
Did my mentioning 60° earlier cause a bell to ring? I hope so! The equilateral triangle has three angles and they all are 60°. If you examine the hexagon in the first picture, you’ll see that it is made from six equilateral triangles. Oh the possibilities! This is why mathematics is important to understand as a person, who is into patchwork.
Now you know how to adjust the height of the WOF strip to any hexagon you want to make in a factory-resembling efficient manner. Kati of from the blue chair has written a nice tutorial on how to make (huge) hexies when there is no template. She’s just using her rotary cutter and ruler.
If you care to watch these little animations on how to draw a hexagon or a pentagon with merely a ruler and a compass, you can create your templates of any size from any piece of paper. Understanding hexagon geometry like this gives you a lot of freedom and endless options.
Well done if you’re still with me! Are you excited about hexies? I am! Feel free to ponder out loud if there’s anything concerning you about this and I’ll see if I can help you. I’m no mathematician, but have done my fair share of practical calculations in varous situations in life.