



 
Carl, I am not sure exactly what you want but it seems to me that the most important aspect is that the logo not be distorted. In your logo the ratio of the height to the width is 0.75/1.2938 = 0.5797. Hence on any size paper if you specify the width then the height will need to be 0.5797 times the width. If the width of the scaled logo is w units and the height is h units then its area is w × h square units. But to maintain the ratio of height to width, h = 0.5797 w. Thus the area is h × w = 0.5797 w^{2}. Hence if you know the area you want, A, then set A = 0.5797 w^{2} and solve for w. If I haven't answered your question write back, Carl wrote back
Carl, I am taking your statement "the size of the logo must relate DIRECTLY to the overall size of the paper or document it is placed on" to mean that the logo must occupy the same proportion of the area of the document as it does on an 8.5 by 11 page. Suppose the logo has dimensions w by h on a page of area a square units and you want to scale it to dimensions W by H on a page of A square units. (In your case w = 1.2938 inches, h = 0.75 inches and a = 8.5 × 11 = 93.5 square inches.) I still claim that you want the ratio of w to h to be the same as the ratio of W to H so there is a constant k so that h = kw and H = kW. Thus the area of the unscaled logo is kw^{2} and the area of the scaled logo is kW^{2}. Hence the unscaled logo occupies kw^{2}/a of the page and the scaled logo occupies kW^{2}/A of its page. If these are the same then
and thus
or
Thus the scale factor is √(A/a) which in your situation is √(A/93.5). Watch out for the units. I think it is easiest to put all the lengths in the same units. Thus in your example of sizing up to 2 feet by 10 feet I would use 24 inches by 120 inches. Thus A = 24 × 120 = 2880 square inches and hence the scale factor is
This will give W in inches. You still may have to fine tune this using your experience as a graphic designer. For a ridiculous example suppose the trade show banner is 4 inch by 720 inches. Its area is still 2880 square inches and the scaled height is 5.550 × 0.75 = 4.16 inches which won't even fit on the banner. I hope this helps,  


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 