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Dividing fabric can be confusing, especially when different terms are used by different rug hookers. Throw in meters vs. yards, and there's a whole other element! Fabric is typically cut off the bolt, by the yard. Rug hooking wool can also vary in width from the bolt. A typical off the bolt yard will measure 36"x57".
It is generally recommended that wool is washed and dried prior to hooking. This fluffs/thickens the wool to make it easier and nicer to work with. Depending on the water temperature, method of drying, dye technique, and other factors, the amount of shrinkage can vary. A typical yard of dyed wool can shrink between 1-4 inches.
Once you have a yard of wool, there are different ways it can be divided. This is where terms like 1/4 yard vs. fat quarter can cause confusion, or 1/2 yard can be two completely different cuts with the same amount of wool.
Wool strips should be cut parallel to the selvedge edge. The 1/8 yard, 1/16 yard and 1/32 yard options are identical when cutting wool using these two methods. There is a difference for 1/4 yard pieces and 1/2 yard pieces. Loopy cuts are the width of the cut strips (16"), with no need to divide and tear the wool along the selvedge. These cuts almost always include the selvedge edge to provide an indication of how the piece should be run through a cutter.
In some cases (for example, doing wool applique), it may be of interest to have a 1/2 yard as more of a square than a rectangle.
Backing is generally divided in a different way. Loopy backing is available in two cuts (1 yard and 1/2 yard sizes), and are cut as follows:
A quick google search of rug hooking fat quarters shows a range of sizes:
15"x28" (420 square inches)16"x26" (416 square inches)17"x23" (391 square inches)
And keep in mind that a fat quarter of wool will be a different amount than a fat quarter of backing. It's best to confirm the actual size of a piece of wool with a vendor when calculating the amount of wool needed for a project.