Consider the following equations:

Where a is area, p is perimeter, and s is side length

We can find the side length with either our area or our perimeter.

Thus, we only need one statment or the other. 

Since ABCD is a square, we just need to know the length of the diagonale to find the length of the side. BE is half the diagonal, therefore knowing its length would help us find the length of the sides.

Statement 1 tells us the length of BE, therefore, with the formula  where  is the diagonal and  the length of side, we can find the length of the side. 

Statement 2 tells us that triangle AEB is isoceles, but it is something we could already have known from the beginning since we are told that E is the midpoint of the diagonal. 

Therefore, statement 1 alone is sufficient.

To find the area of a square we need to find its side length.

In a square, the diagonal allows us to find the other two sides. The diagonal of a square creates two 45/45/90 triangles with special side length ratios.

I) Gives us the diagonal, which we can use to find the side length, which will then help us find the area.

II) Perimeter of a square allows us to find side length, which in turn lets us find area. 

So, either statement is sufficient.

Statement 1) gives the area of the square.  For all positive real numbers, the formula, , or , can be used to find either area or side length interchangeably.

Statement 2) mentions that the diagonal is 1, which is a positive real number.  The formula , can be used to also find the side length.

Either statement alone is sufficient to solve for the length of the square.

Statement 1) mentions that the area of a quadrilateral is 4.  This statement is insufficient to solve for the length of the square because the family of quadrilaterals include any 4-sided shape with 4 interior angles.  Examples of quadrilaterals are squares, rectangles, rhombus, and trapezoids, but the quadrilateral is not necessarily a square.

Statement 2) mentions that all four interior angles of a quadrilateral are right angles.  This narrows down the shape to either a square or a rectangle.  Both shapes have 4 right angles, but there is not enough information to determine if the shape is a square or a rectangle.  

Therefore, neither statement is sufficient to solve for the length of a quadrilateral.

Because all 4 sides of a square are equal, knowing the length of one side is sufficient to answer the question. Using the Pythagorean Theorem, you can calculate the length of 1 side of a square by knowing the length of a diagonal and then calculate the area.

From Statement 1, the radius of the circle can be calculated by working backwards from the area formula, and the diameter can be calculated by doubling this.

From Statement 2, since, by the regularity of the square,  is one fourth of the circle, the length of  can be multiplied by four to get the circumference of the circle. This can be divided by  to obtain this diameter.

This diameter is equal to the the sidelength of the square in which it is inscribed, so it can be squared to obtain the area of the square. This makes each statement alone sufficient to answer the question.

The diameter of the inscribed circle is equal to the the sidelength of the square. From Statement 1, the circumference can be divided by  to obtain this measure, and this can be squared to obtain the area of the square.

Statement 2 gives extraneous information, as, by regularity of the figure, it is already known that  is one fourth of the circle and, subsequently, a  arc.

The area of the black region is one-fourth the difference of the areas of the circle and the square.

If  is the sidelength of the square, then the length of its diagonal - which is also the diameter of the circle - is, by the Pythagorean Theorem, , and the radius . Therefore, if you calculate either the radius or the sidelength, you can calculate the other, allowing you to find the areas of the circle and the square.

Statement 1 allows you to find the sidelength; just divide 40 by 4.

Statement 2 allows you to find the radius; just solve for  in the equation .

Therefore, either one gives you enough information to solve the problem.

You can use the diagonal of a square to find its side length via the ratio of a 45/45/90 triangle.

In this case our triangle will have side lengths of 15.

We can also divide our perimeter by 4 to get our side length, which is again 15. 

Therefore, each statement alone is sufficient to solve the question.