Used Nancys New Geography map. Printed this out with a laser printer and enlarged it to 2.69x with a copy machine. Over all length ended up to be 14.32". Checked the aspect ratio Height/Width before and after the enlargement they were the very close to the same. The rubber balls in the picture below have a diameter of 4.56". Drew a line on the inside of the paper on the exact opposite side of the equator line. Used this to line it up with the seam in the ball so that it would be centered in a north to south direction. Wraped the paper snugly around the ball and taped it so that it did not slip toward the north or south. Made two of them to see if I got the same result, in case the equator slipped. Top and Bottom of both test units.
Notes on the results: The north and south ended up about .25" short of being centered on the geographic pole. The blank spots where you can see the ball do not matter that much. There is no land at these spots, so one can just picture it as water. There is some overlapping of continents or countries especially in the southern hemisphere. This is a concern and indicated a little to big an image for that continent as compared to the diameter of the sphere. There is only so much one can do with flat paper wrapped around a sphere. I think as it stands it gets the point across. One could improve this by making thinner strips but then the appearance in 2D would look like it is cut up. One could possibly adjust the tabs a little however I am not sure how to do this without cutting off valid land masses.