The area of a rectangle is the product of its length and width;

The diagonal divides the square into two 45-45-90 triangles, which have side lengths in the ratio of . Therefore, the diagonal is the side length .

There are two pretty easy ways to solve this problem, both of which require you to see that this is actually a triangle question. 

The first method is the fastest. If you can recognize that the diagonal of a square makes a  triangle, you can then recall that  triangles have side lengths with ratios of . Therefore:

The second way to solve this problem is to use Pythagorean Theorem:

where  and  are leg lengths of a triangle and  the length of the hypotenuse. In this case, we want to solve for the diagonal, which is the hypotenuse of the triangle it forms, so we are solving for :

So again, we get .

By definition, all four sides of a square are the same length. So if one side has a length of , the length and width of the square are both . The diagonal of the square is just the hypotenuse of a right triangle with the length and width as the other two sides, so we can use the Pythagorean Theorem to calculate the length of the diagonal:

From what we are told, we can figure out the side of the square with this simple formula: , where  is the diagonal of the square and  is the length of the side. So the length of the side of our square will be 2. Therefore the radius of the circle will be half that, or 1. Now that we now the radius, we can calculate the area of the circle , where  is the radius of the circle.

To get the area of the square we need to figure out, the length of the side. Which is  possible by calculating the diameter of the circle, which is in fact the diagonal of the square. We can do as follows:

 therefore  the radius is 2 and the diameter is 4. We now know that the diagonal of the square is 4 as well, with the formula , where  is the length of the diagonal and  is the length of the side of the square, we can figure out the length of the side, which is . Therefore, the area is the square of the side or 8. The final ratio that we are looking for is  or .

From the given area of the square we can figure out the length of the side of this square, which is the square root of the area, or 7. Using the formula for the diagonal of the square, , where  is the length of the diagonal, and  the length of the side. Therefore, the length of the diagonal is . In this case since the square is inscribed in the circle, the diameter of the circle is equal to the diagonal of the square. Therefore, we can find the circumference of the circle , or . We just need to divide the result by 2 since we only want the circumference of the semicircle and also add the length of the diagonal, which "closes" the semicircle. The final answer is .

To find the length of the diagonal, you can either use the pythagorean theorem or realize that the diagonal of a square is really the hypotenuse of a right isosceles triangle. Therefore, using the short cut, you know that the diagonal is the side length times . Thus, the answer is: 

To find the perimeter of a square given its area, take the square root of the area to find its sidelength; then, multiply that sidelength by 4.

 is a perfect square trinomial, since 

so its square root is , the sidelength.

Multiply this by 4 to get the perimeter: 

Cutting the triangle in half yields a right triangle with the diagonal becoming the hypotenuse and the other two legs being the sides of the square. Using the Pythagorean Theorem, we can solve for the other legs of the triangle.

Since both sides of the square are equal to eachother, , therefore:

To find the area of the square:

 with leg  being one of the sides