Another primary use of the sector is to quickly find or create a whole number proportional relationship between two dimensions–say the width of a box to its height. Or maybe a book like this one that I happen to have lying around:
Just by eyeballing the form of this book, I suspect that it might be a 2:3 rectangle. In other words, it looks like the height is a half again its width (two halves to three halves is 2:3.) To test my theory, I set the sector so the “6” on each leg touches the side of the book.
If the ratio is indeed to be two to three, the height of the book should come out to the span of the “9”s (because 6:9 is the same as 2:3). So I set my dividers to the nine position on each leg.
What do you know! The height is this span exactly. I bet there is something in that book about this happy harmonious happenstance!