Assume that the length of each midpoint is 1.  This means that the length of each side of the large square is 2, so the area of the larger square is 4 square units.

To find the area of the smaller square, first find the length of each side.  Because the length of each midpoint is 1, each side of the smaller square is  (use either the Pythagorean Theorem or notice that these right trianges are isoceles right trianges, so  can be used).  

The area then of the smaller square is 2 square units.

Comparing the area of the two squares, the larger square is 2 times larger than the smaller square.

A few facts need to be known to solve this problem. Observe that the diagonal of the square cuts it into two right isosceles triangles; therefore, the length of a side of the square to its diagonal is the same as an isosceles right triangle's leg to its hypotenuse: .

Rearrange an solve for .

Now, solve for the area using the formula .

Express the ratio of the two sides of the squares as . The area of each square is one side multiplied by itself, so the ratios of the areas would be . The right side of this equation simplifies to a ratio of .

Since the yard is square in shape, we can divide the perimeter(140ft) by 4, giving us 35ft for each side. We then square 35 to give us the area, 1225 feet. 

The formula for the area of a square is . However, the problem gives us a diagonal and not a side.

Remember that all sides of a square are equal, so the diagonal cuts the square into two equal triangles, each a  right triangle.

If we use the Pythagorean Theorem, we see: 

Plug in our given diagonal to solve.

From here we can plug our answer back into our original equation:

If the area of the half circle is , then the area of a full circle is twice that, or .

Use the formula for the area of a circle to solve for the radius:

36π = πr²

r = 6

If the radius is 6, then the diameter is 12. We know that the sides of the square are the same length as the diameter, so each side has length 12.

Therefore the area of the square is 12 x 12 = 144.

The area of a square is equal to its length times its width, so we need to figure out how long each side of the parking lot is. Since a square has four sides we calculate each side by dividing its perimeter by four.

Each side of the square lot will use 40 feet of fence.

^.

To inscribe means to draw inside a figure so as to touch in as many places as possible without overlapping. The circle is inside the square such that the diameter of the circle is the same as the side of the square, so the side is actually 4 in.  The perimeter of the square = 4s = 4 * 4 = 16 in.

Find the area of Square Y, then calculate the area of Square X.

If the perimeter of Square Y is 24, then each side is 24/4, or 6.

A = 6 * 6 = 36 sq ft, for Square Y

If Square X has 3 times the area, then 3 * 36 = 108 sq ft.

The area of the given square is given by so the side must be 6 in.  The side is reduced by a factor of two, so the new side is 3 in.  The perimeter of the new square is given by .