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.\(\displaystyle A=s^{2}\)

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 \(\displaystyle \sqrt{2}\) (use either the Pythagorean Theorem or notice that these right trianges are isoceles right trianges, so \(\displaystyle s, s, s\sqrt{2}\) 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.

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 perimeter of a square is very easy to calculate. Since all of the sides are the same in length, it is merely:

\(\displaystyle 4s=p\)

\(\displaystyle 4s=50\)

\(\displaystyle s=12.5\)

From this, you can calculate the area merely by squaring the side's value:

\(\displaystyle A = 12.5^2=156.25\) \(\displaystyle \textup{in}^2\)

The area of a square is very easy. You merely need to square the length of any given side. That is, the area is defined as:

\(\displaystyle A=s^2\)

For our data, this is:

\(\displaystyle A=14^2=196\)

For a square all the sides are equal, and there are four sides, so divide the perimeter by 4 to determine the side length. 

\(\displaystyle s = 48/4 = 12\).

Next to find the area of a square, square the side length: 

\(\displaystyle A = s^2 = 12^2 = 144ft^2\).

Don't forget your units!

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.

\(\displaystyle \frac{160}{4}=40\)

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

\(\displaystyle Area=length \times width\)

^{\(\displaystyle 40\ ft\times 40\ ft=1600\ ft^{2}\).}

If a square is inscribed inside a circle, the diameter of the circle is the diagonal of the square. Since we know the radius of the circle is \(\displaystyle 45\) feet, the diameter must be \(\displaystyle 90\) feet. Thus, the diagonal of the square garden is \(\displaystyle 90\) feet.

All squares have congruent sides; thus, the diagonal of a square creates two isosceles right triangles. The ratio of the lengths of the sides of an isosceles right triangle are \(\displaystyle x: x: x\sqrt2\), where \(\displaystyle x\sqrt2\) is the hypotenuse. Thus, to find the length of a side of a square from the diagonal, we must divide by \(\displaystyle \sqrt2\).

\(\displaystyle s = \frac{90}{\sqrt2} = \frac{90\sqrt2}{2} = 45\sqrt2\)

The area of a square is \(\displaystyle A = s^2\), so if one side is \(\displaystyle 45\sqrt2\), our area is

\(\displaystyle A = s^2 = (45\sqrt2)^2 = (2025\cdot 2) = 4050\)

Thus, the area of our square garden is \(\displaystyle 4050ft^2\)

To find area, simply square the side length. Thus,

\(\displaystyle A=1^2=1\)

To solve, simply use the formula for the area of a square given side length s. Thus,

\(\displaystyle A=s^2=5^2=25\)

We know that the radius of the circle is also half the length of the side of the square; therefore, we also know that the length of each side of the square is 12 inches.

We need to square this number to find the area of the square.

\(\displaystyle A=s^2\)

\(\displaystyle A=12^2\)

\(\displaystyle A=144\)