A mathematical scale study that keeps Earth diameter fixed at exactly 100%, extends the earlier E1–M1–I2I–C2C–O2O comparison with an Earth-centred outer circle (EOC), and divides EOC into 30 equal 12° angular sections.
Research · Mathematical Scale Study
This research explores the Earth–Moon distance through a geometric scale model in which the diameter of Earth remains the fixed reference at exactly 100%. The model compares Earth, the Moon, several Earth–Moon distance boundaries, and a larger Earth-centred outer circle while maintaining the same mathematical scale throughout the study.
The purpose of this model is not to create a realistic astronomical illustration or orbital simulation. Instead, it provides a simple mathematical way to visualize very large distances using perfect circles, fixed centres, proportional diameters, and measurable angular sections.
Earth as the Fixed 100% Reference
Earth, identified as E1 in this model, has a diameter of 12,756 km and is defined as exactly 100%. Every other diameter and radius in the study is measured relative to this fixed Earth diameter.
The important rule is that Earth is never resized or renormalized when another circle becomes much larger. E1 always remains 100%, allowing every other measurement to be compared against the same reference.
Earth and Moon Reference Circles
The Moon reference circle, identified as M1, uses a diameter of 3,475 km. Compared with Earth’s 12,756 km diameter, the Moon represents approximately 27.24% of the Earth-diameter reference.
| Circle | Description | Diameter | Diameter % | Radius | Radius % |
|---|---|---|---|---|---|
| M1 | Moon Reference Circle | 3,475 km | 27.24% | 1,737.5 km | 13.62% |
| E1 | Earth Reference Circle | 12,756 km | 100% | 6,378 km | 50% |
| I2I | Inner Boundary to Inner Boundary | 397,384.5 km | 3,115.28% | 198,692.25 km | 1,557.64% |
| C2C | Earth Centre to Moon Centre | 405,500 km | 3,178.90% | 202,750 km | 1,589.45% |
| O2O | Outer Boundary to Outer Boundary | 413,615.5 km | 3,242.52% | 206,807.75 km | 1,621.26% |
| EOC | Earth Outer Circle | 814,475 km | 6,385.03% | 407,237.5 km | 3,192.52% |
The resulting size order is:
M1 < E1 < I2I < C2C < O2O < EOC
Master Scale Formula
The diameter percentage of any circle can be calculated by dividing its physical diameter by the Earth diameter of 12,756 km and multiplying the result by 100.
Diameter % = (Diameter ÷ 12,756 km) × 100
The radius percentage follows the same Earth-diameter reference.
Radius % = (Radius ÷ 12,756 km) × 100
Because the Earth diameter rather than the Earth radius is used as the master reference, the Earth diameter equals 100% while the Earth radius equals 50%.
Earth–Moon Centre-to-Centre Distance
The C2C circle represents the Earth-centre-to-Moon-centre distance used in this model. The fixed modelling distance is 405,500 km.
This centre-to-centre measurement forms the central reference from which the inner and outer Earth–Moon boundary measurements can be calculated.
Inner-to-Inner Distance — I2I
I2I represents the distance between the facing surface of Earth and the facing surface of the Moon. It is calculated by subtracting both the Earth radius and Moon radius from the centre-to-centre distance.
I2I = 405,500 − 6,378 − 1,737.5
I2I = 397,384.5 km
Relative to the fixed Earth diameter, this measurement represents approximately 3,115.28%.
Outer-to-Outer Distance — O2O
O2O represents the complete span from the far outer side of Earth to the far outer side of the Moon. It is calculated by adding the Earth radius and Moon radius to the centre-to-centre distance.
O2O = 6,378 + 405,500 + 1,737.5
O2O = 413,615.5 km
This corresponds to approximately 3,242.52% of the Earth-diameter reference.
Relationship Between I2I, C2C and O2O
The Earth radius and Moon radius together equal 8,115.5 km. Subtracting this amount from C2C produces I2I, while adding the same amount to C2C produces O2O.
- C2C − I2I = 8,115.5 km
- O2O − C2C = 8,115.5 km
- I2I = 397,384.5 km
- C2C = 405,500 km
- O2O = 413,615.5 km
This creates a symmetrical mathematical relationship around the centre-to-centre measurement.
Earth Outer Circle — EOC
The Earth Outer Circle, identified as EOC, introduces a much larger circle whose centre is exactly the same as the centre of Earth E1.
Its radius extends from the centre of Earth to the far outer boundary beyond the Moon. The radius therefore combines the Earth-to-Moon centre distance with the Moon radius.
EOC Radius = 405,500 + 1,737.5
EOC Radius = 407,237.5 km
The complete EOC diameter is twice this radius.
EOC Diameter = 407,237.5 × 2
EOC Diameter = 814,475 km
Compared with the fixed Earth diameter, the EOC radius is approximately 3,192.52% and the complete EOC diameter is approximately 6,385.03%.
EOC Circumference
Once the EOC radius is established, its circumference can be calculated using the standard circumference formula for a circle.
Circumference = 2 × π × Radius
Using an EOC radius of 407,237.5 km gives a circumference of approximately 2,558,748.7 km.
This circumference provides the basis for the next part of the model, where the complete Earth Outer Circle is divided into equal angular sections.
Dividing EOC into 30 Equal Sections
A complete circle contains 360 degrees. Dividing the Earth Outer Circle into exactly 30 equal sections produces an angle of 12 degrees for every section.
360° ÷ 30 = 12°
Each section therefore represents one-thirtieth of the complete EOC circumference.
- Total sections: 30
- Angle per section: 12°
- Share of full circumference: 3.3333%
- Approximate outer arc per section: 85,291.6 km
It is important to distinguish degrees from percentages. Each section has an angular width of 12 degrees, not 12 percent. Thirty sections of 12 degrees each complete the full 360-degree circle.
Earth–Moon EOC 30-Section Scale Diagram
The geometric diagram combines E1, M1, I2I, C2C, O2O and EOC within the same mathematical scale. E1 remains the fixed Earth reference, while EOC is centred directly on Earth and divided into 30 equal radial sections.
Reading the Scale Model
M1 is the smallest reference circle at approximately 27.24% of the Earth diameter, while E1 remains fixed at 100%. The I2I, C2C and O2O circles are each more than thirty-one times the Earth diameter and therefore appear relatively close to one another when compared at this scale.
EOC is considerably larger because its radius alone extends from the centre of Earth to the far outer Moon-side boundary. Once this radius is doubled to form a complete circle, its diameter reaches approximately 6,385.03% of the Earth-diameter reference.
The circles in this model should be interpreted as mathematical representations of selected linear distances. They are not intended to represent the physical orbit of the Moon or actual orbital paths.
Reproducing the Model at Any Size
The same geometry can be reproduced at almost any drawing size because every measurement is expressed relative to Earth.
If Earth E1 is drawn with a diameter of 100 drawing units, the corresponding approximate diameters become:
- M1 = 27.24 units
- E1 = 100 units
- I2I = 3,115.28 units
- C2C = 3,178.90 units
- O2O = 3,242.52 units
- EOC = 6,385.03 units
For the 30-section EOC construction, radial lines can be positioned at 0°, 12°, 24°, 36° and continued at 12-degree intervals until the complete circle contains exactly 30 equal sectors.
Master Size List
| Order | Circle | Diameter % of Earth | Diameter |
|---|---|---|---|
| 1 | M1 | 27.24% | 3,475 km |
| 2 | E1 | 100% | 12,756 km |
| 3 | I2I | 3,115.28% | 397,384.5 km |
| 4 | C2C | 3,178.90% | 405,500 km |
| 5 | O2O | 3,242.52% | 413,615.5 km |
| 6 | EOC | 6,385.03% | 814,475 km |
Scope of the Study
This research is a mathematical scale and geometry study. The 405,500 km Earth-to-Moon centre distance is treated as a fixed modelling value for the purpose of constructing and comparing the circles.
Actual Earth–Moon distance changes because the Moon follows an elliptical orbit. Therefore, this diagram should not be interpreted as stating that the real Earth–Moon separation remains permanently fixed at the value used in this model.
Conclusion
This Earth–Moon scale model provides a consistent method for comparing Earth, the Moon, their selected boundary distances, and a much larger Earth-centred outer circle without changing the original Earth reference.
Earth E1 remains exactly 100%, the Moon M1 remains approximately 27.24%, and the I2I, C2C and O2O measurements show the different ways in which the Earth–Moon span can be represented. The addition of EOC extends the model to a complete Earth-centred circle with a diameter of 814,475 km, equivalent to approximately 6,385.03% of the Earth diameter.
Dividing the EOC into 30 equal sectors creates a precise 12-degree interval for each section. The result is a reproducible geometric framework for studying very large linear distances, proportional circles, radial measurements, angular divisions, and Earth–Moon scale relationships while keeping Earth diameter as the single fixed 100% reference.
Research by Manoj Bhati
www.manojbhati.com
