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Measure the room where you want to install the custom-painted floor and purchase a linoleum remnant larger than those dimensions. Flooring stores typically have larger remnants. Don't worry about the pattern on the linoleum—you'll be painting on the back. But do check the back to make sure it's in good shape, without any cracks or gouges.
Use kraft paper or newspaper to create a template of the floor you're covering. Be sure to cut around any fixtures in the room. Use masking tape around any curved areas. Also, write directly on the template any directions you think you'll need, such as the location of windows or doors.
Note: If the room you're working on has a threshold, allow for extra linoleum so you can run the seam under the threshold. Make sure the template covers the entire area—you don't want the finished piece to come up short.
Using a utility knife or other sharp cutting blade, cut the remnant about 1/8" outside the template. You can always trim the linoleum during installation, but you can't "un-cut" it.
Remove the template and prime the back of the linoleum with a stain-blocking primer. Once the primer is dry, paint the linoleum as desired.
After the paint is dry, brush on three coats of polyurethane. Let each coat dry completely before applying the next one.
Remove the threshold from the room, and adhere double-stick tape in the corners. Install the finished piece. Tuck excess linoleum under baseboards if possible. Otherwise, trim with the utility knife. Replace the threshold and any other trim you removed for the installation.