The diagonal of a square creates two special 45-45-90 triangles, meaning that the diagonal of a square is just the length of one side of the square multiplied by the square root of 2.  

In this problem, you can figure out the length of one side of the square by finding the square root of the area (which is equal to a side length), then multiplying that number by the square root of 2.

Assign variables such that

One side of ABCD = a

and One side of EFGH = e

Note that all sides are the same in a square. Since the perimeter is the sum of all sides, according to the question:

4a = 3 x 4e = 12e or a = 3e

From that area of EFGH is 25,

e x e = 25 so e = 5

Substitute a = 3e so a = 15

We aren’t done. Since we were asked for the area of ABCD, this is a x a = 225.

Let S be the original side length. S*S would represent the original area. Doubling the side length would give you 2S*2S, simplifying to 4*(S*S), giving a new area of 4x the original, or 144.

Area of a square in terms of each of its sides:

  Area = S x S

Perimeter of a square:

  Perimeter = 4S

So if 'the perimeter of a square is equal to twice its area':

  2 x Area = Perimeter

  2 x [S x S] = [4S]; divide by 2:

  S x S = 2S; divide by S:

  S = 2

If Freddie uses x feet of fencing makes a square, each side must be x/4 feet long. The area of this square is (x/4)² = x²/16 = p square feet.

If Freddie uses one third of x feet of fencing makes a square, each side must be x/12 feet long. The area of this square is (x/12)² = x²/144 = 1/9(x²/16) = 1/9(p) = p/9 square feet.

Alternate method:

The scale factor between the small perimeter and the larger perimeter = 1 : 3. Since we're comparing area, a two-dimensional measurement, we can square the scale factor and see that the ratio of the areas is 1² : 3² = 1 : 9.

This is an isosceles right triangle, so the diagonal must equal times the length of a side. Thus, one side of the square measures , and the area is equal to 

The area of  is , the area of  is . Therefore, you can fit 5.06  in .

The perimeter of a square is given by  so the side length of the original square is   The side of the new square is enlarged by a factor of 3 to give  

So the area of the new square is given by .

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.

We start by dividing the area of square R (48) by 12, to come up with the area of square T, 4.  Then take the square root of the area to get the length of one side, giving us 2.