# Theorem 1. (Golden ratio of Earth Square and Circle)

The ratio of the area of the Earth Square to the area of the Earth Circle reflects the golden ratio.

#### Proof.

Let there be a square with a side length of 11 (see Image 35. right). Now the area of the square is formed,

*A _{square }= 11² = 121,*

and also the area of the circle

*A _{circle }= π * r^{2 }= π * (5,5)^{2 }≈ 95,033.*

Thus, there is

*A _{square }/ A_{circle} = 11² / (π * (5,5)^{2} ) ≈ 1,273 ≈ √φ.*

Therefore the ratio of the area of the Earth Square (also Mona Lisa Square) to the area of the Earth Circle reflects approximation of the golden ratio.

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## Theorem 2. (Squaring the Earth-Moon circle)

The perimeter of the Earth Square equals the circumference of the circle with the Earth-Moon radius.^{27}

^{27} Squaring the circle usually means that either 1) that the perimeter of the square and the circumference of the circle are of equal length (equal lines), or 2) the areas of the circle and the square are equal (equal areas).

#### Proof.

Now, the perimeter p_{1} of the Earth Square is

*p _{1 }= 5,5 * 8 = 11 * 4 = 44.*

Correspondingly, the perimeter p_{2} of the circle with the Earth-Moon radius is

*p _{2 }= 2πr = 14π ≈ 43,98 ≈ 44.*

Now, we have an approximation 44 and therefore

*p _{1} = p_{2}.*

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Notice: If we choose to use the Ancient value of π (22/7), Theorem 2 gives us exactly the same value. This is a simple and beautiful equation:

*p _{2 }= 2 * (22 / 7) * 7 = 44 = p_{1}.*

## Theorem 3. (Golden ratio of Earth and Moon radiuses)

The ratio of the radius of the Moon to the radius of the Earth reflects the golden ratio.

#### Proof.

Using the values of Image 36, we now have

*r _{moon }/ r_{earth} = 1,5 / 5,5 ≈ 0,273 ≈ √(φ - 1).*

So the ratio of the radius of the Moon to the radius of the Earth reflects the golden ratio.

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