To find the ratio of the areas, we are obviously going to first need to find the areas of the inner and outer squares. The outer square has a side length of because we are units away from 0 on the x-axis. Thus, the area of the outer square is .
We can see that a right triangle is formed by part of the x-axis, part of the y-axis, and the side of the inner square. Thus, we can find the length of the inner square through the Pythagorean Theorem, which is , where and are the legs of the right triangle and is the hypotenuse. The lengths of the legs of the right triangle in the picture are and . We can use the Pythagorean Theorem to find the other side length.
We have now found that the side length of the inner square is . Thus, the area of the inner square is .
Using the two areas we just found, we can say that the ratio of the area of the inner square to the area of the outer square is , or choice A.