9514 1404 393
Answer:
3 -2√2
Explanation:
For a regular octagon of side length s, the "flat-to-flat" dimension across the center is s(1+√2). The flat-to-flat dimension of the shaded center octagon is simply s. The ratio of areas is the square of the ratio of linear dimensions, so we have ...
small area / large area = (1/(1+√2))^2 = 1/(3+2√2)
small area/large area = 3 -2√2
_____
The attachment shows octagons after the fashion described here. The side length of the larger one is 2 units. The numbers inside are the areas. They have the ratio shown above.