"The ratio of the areas of the two polygons is the square of the ratio of the sides. So if the sides are in the ratio 3:1 then the areas will be in the ratio 9:1." In this case, the ratio of the two regular octagons is 50:18, or 25:9. Thus, the ratio of the sides is 5:3.
To see that the ratio of the sides is also the ratio of the perimeters and is, therefore, also 5:3, say the small octagon has side length x, then its perimeter is 8x. Now, the big octagon has length side (5/3)x, and perimeter 8*(5/3)*x. The ratio of the perimeters of big octagon to the small is [8*(5/3)]:8 or just (5/3):1, equivalent to 5:3.
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