I toss out a lot of math in my quilting blogs. But when you think about quilting, there’s a great deal of geometric work involved. I’ve always taken the math end of the art for granted until a couple of weeks ago when I went to get my oil changed. In North Carolina the Delta variant numbers are still pretty high. The oil-changing establishment I went to closed their small waiting room and were allowing everyone to stay in their car while oil is changed and fluids are checked. So, while I was waiting for the mechanics to take care of my Tahoe, I read a quilting book. The lead mechanic noticed.
“Are you a quilter?” he asked.
I nodded. “Over 30 years.”
“My wife just started quilting. She made quilts for Christmas presents. Quilters are engineers! There’s so much math and planning involved!”
I never thought of quilters are engineers, but I guess we are to a degree. But mathematicians? We’ve been mathing out quilts for generations.
As a whole, quilters can read and follow pattern directions pretty well. However, altering the pattern, reproducing a quilt without a pattern (for example, making a copy of an antique quilt), or designing your own quilt can be a challenge if you don’t know how to “math” the quilt out. There are formulas for just about everything. And instead of this freeing up quilters to make the quilt they want, I find most quilters are just a bit intimidated by the math. Don’t be. I will tell you what I told the countless chemistry and physic students who passed through my classroom: Numbers are your friends. They don’t lie.
The great thing about quilt formulas is they never change. They’re pretty stable across the board. Over the years I’ve mentioned lots of formulas. Finally, a good quilting buddy of mine asked if I would put all the formulas in one blog post so she wouldn’t have to Google each formula when she needed to use it. This seemed like a good idea, so the here we go…
Before we take a deep dive into block units, there are two important facts to remember:
- When adding seam allowances, etc. to the unit, use the finished size. For instance, if we want to reproduce a finished 2-inch square, we begin with 2-inches.
- These formulas work only with ¼-seam allowances.
Squares and Rectangles
Finished size + ½-inch. For example, if I need a 3-inch finished size, I will cut a 3 ½-inch square of fabric. The added ½-inch allows for ¼-inch seam allowances on all sides. If I needed a 3-inch x 5-inch finished rectangle, I would add ½-inch to both measurements and cut a 3 ½-inch x 5 ½-inch unfinished rectangle.
This formula assumes the length needed is WOF. For this, you simply add ½-inch to the needed width. If you need a 2-inch strip WOF, add ½-inch and cut the strip at 2 ½-inches by WOF (width of fabric).
This formula works whether you’re constructing the HSTs by placing two pieces of fabric right sides together, drawing a diagonal line from one corner to another, and then sewing two seams, ¼-inch away from both sides of the diagonal line, or simply cutting a square on the diagonal to make two triangles.
Finished size of HST + 7/8.
Keep in mind each HST unit made this way actually produces two HSTs.
This block unit involves four triangles and most of the time this unit will involve cutting four squares of fabric – meaning each square can produce four QSTs, depending on if each fabric is only used once. For this formula, take the finished size of the QST and add 1 ¼-inches. Cut your squares this size, and then cut each of them twice on the diagonal.
Finished size of QST = 4-inches.
4-inches + 1 ¼-inches = 5 ¼-inches.
Cut four squares (one in each fabric needed) and then cut each square twice on the diagonal.
A trapezoid is a rectangle which has each end cut on a 45-degree angle in opposite directions.
This block unit works a little differently than the prior ones. For this shape, take the finished height and add ½-inch. So, if you need the trapezoid 3-inches in height, you will cut your fabric strip at 3 ½-inches. The length works a little differently. Since you must cut off both ends of the strip at 45-degree angles, you have to add more than just the standard ½-inch for the seam allowance. For the trapezoid’s unfinished length, add 1 ¼-inches to the finished length. If we need a 7-inch finished trapezoid in length, we will add 1 ¼-inch and cut our strip 8 ¼-inches long.
So, for a 3-inch x 7-inch trapezoid, we will cut a strip of fabric 3 ½-inches x 8 ¼-inches. Then we align the 45-degree angle on a ruler with the horizontal edge of the rectangle and carefully cut the end of the strip from the bottom corner to the top edge. Flip the trapezoid upside down (or rotate the mat). Align the 45-degree angle on the ruler with the edge of the shape and cut again.
Occasionally you will see this in a quilt block:
This is a half-trapezoid – only one of the ends has been cut at the 45-degree angle. This isn’t any more difficult to make than a trapezoid, but the math is a little different. The height is determined the same way – add ½-inch to the finished height. However, for the length you only need to add 7/8-inch. Using the same trapezoid finished measurements as above (3-inches x 7-inches), we will cut the unfinished trapezoid 3 ½-inches x 7 7/8-inches. Then simply cut the whichever end needs to be slanted at 45-degree angle.
Isosceles Triangle in a Square
This unit involves cutting two shapes: background triangles which are mirror images of each other and the isosceles triangle itself. It’s also important to remember the sides of the isosceles triangle are of a different measurement than the base of the triangle. The background of this block has a left- and right-facing sides (the mirror images). In order to cut the fabric correctly for this, we will need to cut two rectangular pieces of fabric with their right sides together. This will give us the mirror images needed. To get the correct size rectangles needed to do this, take the desired finished size of base (this will be the shortest side of the mirrored triangle) of the mirror image triangle and add ¼-inches. For instance, if this number is 3-inches, add 3/4-inch to this and cut a strip of fabric 3 3/4-inch long, by the WOF.
To determine the length, take the desired height of the block and add 1 ¼-inches. So, if our finished height is 5-inches, we will cut the 3 ¼-inch strip into 6 ¼- sections. Place two of these sections right sides together and make a diagonal cut from one corner to the other. We will need to make an equal number of diagonal cuts from left bottom corner to the right top corner (these are the “lefties”) and an equal number from the top left corner to the bottom right corner (these are the “righties”).
Now for the triangle in the middle. The definition of an isosceles triangle tells us the two sides of the triangle are the same length, but the base (bottom) of the triangle is different. And it’s the base we will work with first. In order to get the width of the isosceles triangle, take the finished base measurement and add 7/8-inch to it. So, if the base of our triangle is 3-inches, we will cut a WOF strip 3 7/8-inch wide. To figure the height of the triangle, we do the exact same thing. We take the finished height and add 7/8-inch to it. If the desired height is 2 ½-inches, we add 7/8-inch to that and know we need to sub-cut the 3 7/8-inch WOF strip into 3 3/8-inch rectangles.
Cutting the triangles from these rectangles is super-easy. Fold the rectangle in half to find the center and finger press a crease into the fabric. Line a ruler up with one of the lower corners and allow it to meet the center at the fold at the top of the rectangle. Cut. Repeat the process on the other side of the fabric.
Word of wisdom here – for me, this method works well if I’m only cutting a few isosceles triangles in squares. However, remember this quilt?
It felt as if I had thousands of isosceles triangles in squares in this quilt. Instead of cutting, sub-cutting, and creasing the fabrics, I purchased this little jewel:
I don’t often purchase specialty rulers, but for me, these paid for themselves with this quilt. These rulers put all the moving parts of the isosceles triangle in a template, so I didn’t have to fret over the math. Two rulers solved any issues. I simply cut a strip of fabric (WOF) the width the instructions directed, and then used the ruler templates at the appropriate markings to make my skinny side triangles and the center isosceles triangle. This saved me sooooooo much time!
The sides and base on an equilateral triangle are all the same measurement. And like the isosceles triangle, this begins with a strip. Take the finished height of the equilateral triangle and add ¾-inch. For instance, if the finished height of the equilateral triangle needs to be 5-inches, we need to cut a strip of fabric 5 ¾-inches wide x WOF.
Align the 60-degree mark on a ruler with the bottom edge of the strip and make the first cut. Discard this piece. Rotate the ruler so the other 60-degree marking is aligned with the top edge of the strip and cut. Before making any additional triangles, verify that the measurements of the triangle from to top bottom is the measurement of the strip. If there are any errors – such as the 60-degree cut is off – it will show up at this point. If all is well, continue rotating your ruler in this manner until you’ve cut all the triangles you can out of the strip of fabric. If there is an error, correct it before continuing to cut out the triangles.
If you find yourself cutting a lot of equilateral triangles, there is a ruler:
I’ll be upfront here and let you know there are hundreds of 45-degree diamond rulers on the market. And if I were constructing a quilt with a lot of them, I’d look into one of these rulers. My favorite diamond ruler is this one:
This little jewel allows you to cut diamonds, equilateral triangles, and triangles. But if you’re only cutting a few diamonds every now and again, you may want to keep the formula in mind.
Diamonds, like triangles, begin with a strip of fabric. Take the desired finished width of the diamond and ½-inch. This is how wide the WOF strip needs to be. For instance, if we need a 3 ½-inch wide finished diamond, we simply add ½-inch and make our WOF strip 4-inches wide. Place the strip in a horizontal position and square off the end. Align the 45-degree marking on the ruler with the top horizontal edge of the strip and cut. Move the ruler across the strip to the appropriate measurement (in this case 4-inches), making sure to keep the 45-degree marking aligned with the edge of the strip. Make the second cut and repeat until you have made all the diamonds you need.
If the diamond is elongated, the piece will have a left and right side – just like the isosceles triangle in a square. The images will mirror each other. If this is the case, cut a strip the finished width, plus ½-inch across WOF. Fold the strip right sides together and cut, keeping the 45-degree mark aligned with the top of the strip.
These are the formulas for the most commonly used block units. In the second half of this post, we will review the Golden Ratio and Quilter’s Cake. This will be a brief review, as I’ve discussed both of these at length, but the request was I put all my quilting math formulas in one place.
The Golden Ratio
1.618 — this is the number that’s “golden.” While this ratio has been used in everything from art to zoology, quilters use it primarily for sashing and borders. It works like this:
1. To determine how wide a sashing can be, multiply the size of the finished block by 1.618 and then divide by 4. For instance, if the finished block size is 8-inches, it would work like this:
8 x 1.618 = 13
13/4 = 3 ¼-inches. The widest the sashing could be and still look balanced against the block is 3 ¼-inches (finished).
To determine how narrow the sashing cand be, multiply by roughly half the Golden Ratio (.618) and then divide by 4 again. If our finished block size is 8-inches, we would calculate the narrow sashing like this:
8 x .618 = 5
5/4 = 1 ¼-inches. The narrowest the sashing could be and still look balanced is 1 ¼-inches.
You also need to remember the width of the sashing can be anywhere between 1 ¼-inches and 3 ¼-inches and it will look just fine. Anything narrower than the smallest number or bigger than the largest number will look wonky.
We also use the Golden Ratio for estimating our borders. To do this, we have to take the size of the finished block + the sashing. Using our 8-inch block from the above examples, let’s say we sewed 2-inch sashing to the block.
8-inch block + 2-inch sashing = 10-inches. (We always work with finished numbers and then add seam allowances)
10-inches x 1.618 = 16.18
Then divide by 4, since there are four sides on a quilt.
16.18/4 = 4-inches
The widest the borders need to be is 4-inches.
For the how narrow the borders can be, we take the block size + the sashing x .618
Using the example above, the math would look like this:
8-inch block + 2-inch sashing = 10-inches
10 x .618 = 6.18
6.18/4 = 1 ½-inches
The narrowest the borders need to be is 1 ½-inches
Now to get to the total of the widest possible border, we can split that border up into multiple borders of varying widths until the sum of the borders equals the largest width. So, using the above example with a 4-inch border, we could have two borders at 2-inches each, two borders with one 3-inches and the other 1-inch, or three borders each 1 1/3-inches wide — or any other variation which will total 4-inches.
This number is 1.414. I call this “Quilter’s Cake” because it makes the formulas fun
and otherwise it’s just one of those geometry numbers used to figure out triangles with two 45-degree angles and one 90-degree angle –which is exactly what we’re doing with on-point settings, but no one wants to remember their high school geometry class while they’re quilting. Quilter’s Cake is used in on-point quilt settings such as the one below.
There are side triangles and corner triangles. The number of side triangles will depend on the number of rows in the quilt – the more rows the more side triangles needed. However, there are always only four corner triangles, because most quilts only have four corners. These are the triangles we’ll deal with first.
Corner Triangles: Take the finished size of the block, divide it by 1.414, and then add 7/8-inch seam allowance. So still using our 8-inch finished block, the math will look like this:
8/1.414 = 5 2/3
5 2/3 + 7/8 = 6 ½.
We will need two 6 ½-inch squares, cut on the diagonal to make the four corner triangles.
Side Triangles: For the sake of example, let’s say our quilt has ten side triangles – three on the right side, three on the left side, two along the top, and two along the bottom. This time we multiply by 1.414 and add 1 ¼-inch seam allowance.
Still using our 8-inch blocks, the math works this way:
8 x 1.414 = 11 1/3-inches
11 1/3 + 1 ¼ = 12 ½
These squares are cut twice on the diagonal, so we get four triangles per square.
Since we need ten triangles: 10/4 = 2.5, which we will round up to three. We will need to cut three 12 ½-inch squares and then cut them twice on the diagonal.
There you go…all my quilter’s math in one blog. There are additional formulas for estimating yardage, however they are all together in the following blogs:
If you want information on that, go there.
I hope this keeps everyone from Googling for hours to find my quilter’s math!
Now a quick update on my brother, Eric. Many of you have asked how he’s doing. I am so happy, thankful, and joyful to report the stem cell transplant is over. His body responded well to the procedure and he left the step-down unit yesterday and went home. His numbers are well within the normal range, and other than his hemoglobin being a little low (which is to be expected) and his liver enzymes are off (due to the meds), he’s doing very well. He will return to meet and talk with his post-SCT team on Monday, but doesn’t have to return to UNC for labs until Nov. 30. He can’t be around people because his immune system is still compromised. I’m just so glad this is behind him and he’s home. Continue to keep him in your prayers — and thank you for praying for him!
Until next week, Quilt On!
Love and Stitches,
Sherri and Sam