Quilting: The Mariner's Compass - March 11-15, 1996
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I made up this problem in honor of Pi Day, which I believe is this Wednesday, March 13, and National Quilt Day, which is this Friday, March 15.There is a quilt pattern called the Mariner's Compass. It's a hard pattern to sew because it's made of lots of little pieces all built in a circular design (way too complicated for me - I stick to squares and rectangles). There are two versions of it, and we will look at the one that's less familiar but easier to describe.
The pattern starts with a circle. Around the circle are isosceles triangles, with their bases on the circle. Then there is another row of isosceles triangles going around the first, and that pattern continues.
It is often hard for quilters to figure out how much fabric they will need before they start, so we're going to figure out how much you might need for a queen-size quilt.
In this quilt, the center circle (which is just a solid color) is 11 inches in diameter. The isosceles triangles have 4-inch bases and are 6 inches high, so the circles are 6 inches apart. On each layer, put as many triangles as will fit around the circle (you can't have part of a triangle). Go out enough layers so that the quilt is as wide as it can be without being over 90 inches (that's about the size of a queen-size quilt).
How many layers are there? How many triangles?
Now let's figure out how many yards of fabric that is. When sewing quilt pieces together, each piece has a 1/4-inch seam. That would make our triangles about 4.6 inches at the base and 7 inches tall when we cut them out. Fabric is usually 44 inches wide. Figure out how many yards of fabric we'll need to make all the triangles. Round up to the nearest quarter yard, because you always want some extra just in case!
Extra: We're going to use plain white fabric between the triangles. Can you figure out about how much of white fabric we would need to fill out the 90-inch square?
Solutions
Figuring out patterns and how much fabric you need for a quilt is a hard thing to do - many quilters buy books that explain to them exactly how much of everything to buy and how to cut everything, because they can't or don't want to try to figure it out. This proved a slightly difficult task for all of you as well.
A number of people went one layer too far, forgetting that when you add 12 inches for the circle, you then still have another 12 inches used by the triangles that you stick on that layer.
Finding the amount of fabric was interesting. If you just multiply the number of triangles (190) by the area of each one, you get just under 2 yards. But that doesn't take into account the fact that you need to be able to fit each whole triangle onto the fabric - it's not just a straight area conversion. A couple of folks did the straight area thing, and I gave it to them because that's a tough step to realize. But a couple of other folks did an excellent job - check out the diagrams by Nathan and Thomas!
Some folks came up with answers for the area of the little triangles that were greater than the area of the whole quilt! You need to be able to look at your answer and decide whether it's reasonable or not. In a 90-inch by 90-inch quilt, that's about 3 yards by 3 yards, so if you come up with an answer for the triangles that's greater than 10 yards, there's something wrong.
Following are highlights. The names of all the people who submitted correct solutions and most of the solutions are also available - and a big welcome to our first school from Louisiana! A few folks did the extra part as well - deciding how much white fabric it would take. Take a look at these and see if you can decide how well they did.
Thomas S. Kuo
Grade: 7
School: Murray Junior High School, Ridgecrest, California
1. 6 layers make the quilt under 90". I got this by making a table,
as shown in the following:
# layer | Diameter of circle | Cross Section | # of triangle
| (in inches) | (includes circle |
| | and triangles) |
----------------------------------------------------------------
1 | 11 | 23 | 11*pi/4 ~ 8
2 | 23 | 35 | 23*pi/4 ~ 18
3 | 35 | 47 | 47*pi/4 ~ 27
4 | 47 | 59 | 59*pi/4 ~ 36
5 | 59 | 71 | 71*pi/4 ~ 46
6 | 71 | 83 | 83*pi/4 ~ 55
7 | 83 | 95 | out of range 90"
2. From the above table, I got the total number of triangles 190.
3. A total of two and one quarter yards of fabric is needed for the
triangles.
* * * * * * * * * * * * * . . . . . . * * * * * * * * E
* * * * * * * * *.
* * * * * * * * * .
* * * * * * * * * .
* * * * * * * * * .
* * * * * * * * * * * * * * * *. . . . . . * * * * * *. . .
A B C D
(1) AB = 4.6", ED = 7", and CD = 2.3".
Let n be the number of triangles along the bottom (there are
another n triangles along the top).
Solve equation 4.6 n + 2.3 <= 44 for n
n <= 9.06 and n = 9 for an integer.
(2) There are therefore 18 triangles in a piece of fabric of size
44" x 7".
(3) To determine the length of fabric, I have to divide 190 by 18,
which is 10.56, which means I need 11 pieces of fabric of size
44" x 7" to make the 190 triangles.
(4) I take the 11 pieces and multiply it by 7" which is 77" in total
or 2.14 yards which rounds up to 2 and 1 quarter yards.
EXTRA:
Four and half square yards white fabric are needed to fill out the
90" square.
I got it with:
90 * 90 - 1/2 * 4 * 6 * 190 = 5820 square inches = 4.49 square
yards
Approximately 4 1/2 square yards
Nathan Jacobson
Grade: 6
School: Episcopal School of Acadiana, Cade, Louisiana
Answer:
How many layers? 7 layers
How many triangles? 255 triangles
How much cloth would we need for the triangles? 3 yards
Here is how I got my answer:
Layer | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
------------------------------------------------------------
Diameter | 11" | 23" | 35" | 47" | 59" | 71" | 83" |
of circle | | | | | | | |
------------------------------------------------------------
No. of Tri. | 8 | 18 | 27 | 36 | 46 | 55 | 65 |
Layers:7 layers
Triangles:8+18+27+36+46+55+65=255 triangles
How much cloth for the triangles: 3 yards
[Nathan then corrected his solution]
I realize what I did wrong with my chart and found out that there
are 90 triangles and 6 layers. Now I need only 2.25 yards for the
triangles.
Here is how I figured it out.
/\/\/\/\/\/\/\/\/\
/\/\/\/\/\/\/\/\/\
/\/\/\/\/\/\/\/\/\
Each row has 18 triangles (9 in one direction /\ and 9 in the
other \/) and is 7" high.
(190/18) X (7/36 yards) = 2.05 -- Buy 2.25 yards.
David Trail
Grade 9
Lincoln-Sudbury High School
Number of Layers = 11" + 12" * n * 90"
n * 6
Layers = 6
Number of Triangles = (*d)/4
Layer 1 = (** 11)/4 = 8
Layer 2 = (** 23)/4 = 18
Layer 3 = (** 35)/4 = 27
Layer 4 = (**47)/4 = 36
Layer 5 = (** 59)/4 = 46
Layer 6 = (** 71)/4 = 55
8 + 18 + 27 + 36 + 46 + 55 = 190
190 triangles
Length of Fabric
(1/2*4.6)7
2.3*7=16.1
190 * 16.1 = 3059
44x = 3059
x = 69.5
69.5/36 * 2 yrds.
Amount of plain white fabric
*(5.5*5.5)
+ 3059
3059 90 8100
+ 95 *90 -3154
3154 8100 4946
4946 Square Inches of white fabric
Claire Bonner Stephanie Miesnik Mt. St. Joseph's Academy Grade 10 If the original circle were 11 inches, the next larger circle would have to be 23 inches. It would have to be a foot larger because the triangles go on all sides of the circle. That means it would be six inches on every side. The amount of diameter increase would be constant because the height of the triangles is constant. The other diameters would be 35, 47, 59, and 71 inches. To figure out how many triangles surrounded each circle, we used the base of 4 inches and the circumference of each circle. We would find the circumference and then divide by the base of 4. We realized that we must round down to make sure there would be enough room for a full triangle. The total number of circles was 190. There were 8 triangles around the 11 in. circle, 18 triangles around the 23in. circle, 36 triangles around the 35in. circle, 46 triangles around the 47in circle, and finally 55 triangles around the 71in circle. The reason the largest diameter is only 71in. is the next larger circle would have been 83in. If an 83in. circle were surrounded by triangles, the width would go over the 90 inch limit. To find out how much fabric to order, we calculated the the area of one triangle then multiplied it by 190, the number of triangles. The area of all the triangles was 3059 inches squared. With the width of the fabric being 44 in., we knew to divide 3059 by 44. We then converted our calculation to yards. The length of the fabric should be 2 yds. This amount will allow for a little extra fabric just in case.
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