Kepler's Third Law and The Astronomical Distance

Kepler leaned heavily on Tycho Brahe's naked-eye positional data to formulate his third law -- and astronomers of the same era used pure geometry and trigonometry to measure the actual distances from the Sun to the inner planets, expressed in units of the Earth-Sun distance itself (the astronomical unit, or AU).

Measuring Venus's and Mercury's distance from the Sun. Since Venus and Mercury orbit inside Earth's orbit, there is a maximum angle they can ever appear away from the Sun as seen from Earth -- called the maximum elongation -- which is about $46^\circ$ for Venus and $22.5^\circ$ for Mercury. At exactly this maximum elongation, the Earth-Venus-Sun triangle happens to have a right angle at Venus (Venus-Sun line is perpendicular to the Venus-Earth line), which lets simple right-triangle trigonometry give the Sun-Venus distance directly:

so $r = R\sin\theta$. For Venus, $\sin46^\circ\approx0.72$, giving a Sun-Venus distance of about $0.72\ \text{AU}$; the same method for Mercury ($\theta=22.5^\circ$) gives about $0.38\ \text{AU}$. (Exterior planets like Mars and Jupiter, which have no maximum-elongation limit, need a slightly different geometric method not detailed here.) …