We need to use the Pythagorean Theorem in order to solve this problem. We can write:

where is the diagonal length and is the side length. The diagonal length of the square is , so we can write:

Start by working backwards from the area of the circle to find its diameter.

The area of a circle is , and  (you can get this by trial and error pretty quickly, since you know it's between 3 and 4, and since 12.25 ends in a 5, you now that a 5 is involved), so the diameter of the circle is 7. This is also the diagonal of the square. Y

ou may remember that the legs of a  right triangle (of which we have two within the square, once we draw the diagonal) are equal to the hypotenuse divided by . But even if you forget this, you should recall the Pythagorean Theorem that states that . In this case,  and  are equal, so the square of each side is equal to half of , or 24.5. And the square of one side of a square is also equal to the area of the square.

Use the perimeter to find the side length of the square.

Now, use the side length to find the area of the square.

First, use the perimeter to find the side lengths of the square.

Use this information to find the area.

Since we know the lengths of one leg and the hypotenuse, we can calculate the length of the other leg using the Pythagorean Theorem. We can use this form:

Setting  and  equal to the lengths of the hypotenuse and the leg - 13 and 5, respectively:

The perimeter is equal to the sum of the lengths of the three sides:

This is also the perimeter of the square, so the length of each side is one fourth of this, or

The area is the square of this, or

First, we need the radius  of the circle, which can be determined from the area of a circle formula by setting :

Simplifying the expression by splitting the radicand and rationalizing the denominator:

The circumference of the circle is  multiplied by the radius, or

This is also the perimeter of the square, so the length of each side is one fourth of this perimeter, or

The area of the square is the square of this common sidelength, or 

.

A circle with diameter 10 has radius half this, or 5. The area of the circle can be found using the formula

,

setting :

.

The square also has this area. The length of one side of this square is equal to the square root of this, which is

Simplify this by breaking the radicand, as follows:

The area of a right triangle is equal to half the product of the lengths of its legs. Since we know the lengths of one leg and the hypotenuse, we can calculate the length of the other leg using the Pythagorean Theorem. We can use this form:

Setting  and  equal to the lengths of the hypotenuse and the known leg - 13 and 5, respectively:

The area of the above triangle is

The square also has this area. The length of one side of this square is equal to the square root of this, which is .

A rectangle has perimeter ,  the length and  the width. Substitute  and  in the perimeter formula, and simplify.

In inches, the perimeter of the rectangle can be calculated by substituting  in the following formula:

The perimeter is 110 inches.

Now divide by 12 to convert to feet:

This makes the perimeter 9 feet 2 inches, which is between 9 feet and 10 feet.