1. The problem is to understand and prove why the radius of the Earth is calculated as it is. 2. One historical method, used by Eratosthenes, to estimate Earth's radius involves measuring the angle of the shadow cast by a stick (gnomon) in two different locations at the same time. 3. Eratosthenes knew that at noon during the summer solstice in Syene (now Aswan, Egypt), the sun was directly overhead and shadows were minimal. 4. In Alexandria, north of Syene, he measured the shadow angle as approximately 7.2 degrees, or $\frac{1}{50}$ of a full circle ($360^{\circ}$). 5. Assuming the Earth is spherical, the arc distance between the two points represents $\frac{1}{50}$ of Earth's full circumference. 6. He then measured the linear distance between Syene and Alexandria to be about 5000 stadia (ancient unit), so the full circumference $C$ of Earth is calculated as: $$C = 5000 \times 50 = 250,000 \text{ stadia}$$ 7. Radius $r$ relates to circumference through the formula: $$C = 2\pi r \Rightarrow r = \frac{C}{2\pi}$$ 8. Substituting the circumference into the formula, we get: $$r = \frac{250,000}{2\pi} \approx 39,788 \text{ stadia}$$ 9. In modern units, converting stadia to kilometers gives a radius near 6,371 km, which agrees well with current measurements. 10. This method shows how geometrical observations and assumptions about Earth's shape allowed early scientists to calculate its radius accurately. The Earth’s radius was calculated by using simple geometry, measured shadow angles, and an estimated distance between two points, proving the Earth is spherical and allowing calculation of its size.