Parallax and the sizes of bodies

If the angular radius ρ and distance D of a body are known, the linear radius is r = D · sin ρ ≈ D · ρ (ρ in radians). The Sun and Moon have angular diameters ≈ 0.5° (Sun ≈ 32′, Moon ≈ 31′), so the Sun’s radius is ≈ 696,000 km (≈ 109 times the Earth’s) and the Moon’s ≈ 1737 km (≈ 0.27 of the Earth’s). Eratosthenes (3rd century BCE) found the Earth’s radius from the distance between Alexandria and Syene and the difference in the Sun’s angle. Biruni (early 11th century, on the hill of Nandana fort near Pind Dadan Khan in present-day Pakistan) measured the height of a mountain and the dip of the horizon and found an Earth radius of ≈ 6340 km – very close to today’s mean value, 6371 km. In Biruni’s method, from a mountain’s height h and the horizon’s dip angle θ, R = h · cos θ / (1 – cos θ).