20.2: The Distance to the Moon

The points AB are also located on another circle, centered on the Moon. The radius in that case is the distance \(R\) to the Moon, and because the arc AB covers 0.1 degrees, we get \[AB\,=\,\frac{2\pi r}{360}\times 0.1\]

Strictly speaking, each of the two arcs AB expressed in the above equations is measured along a different circle, with a different radius (and the two circles curve in opposite ways). However, in both cases AB covers only a small part of the circle, so that as an approximation we may regard each of the arcs as equal to the straight-line distance AB. That assumption allows us to regard the two expressions as equal and to write \[\frac{2\pi r}{360}\times 0.1\,=\,\frac{2\pi r}{360}\times 9\]

Multiplying both sides by 360 and dividing by \(2\pi\) give \[0.1R\,=\,9r\,\Rightarrow\,\frac{R}{r}\,=\,90\] suggesting the Moon's distance is 90 Earth radii, an overestimate of about 50%.

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