Distance Between Points of Same Latitude along Parallel of Latitude

Theorem

Let $J$ and $K$ be points on Earth's surface that have the same latitude.

Let $JK$ be the length of the arc joining $JK$ measured along the parallel of latitude on which they both lie.

Let $R$ denote the center of the parallel of latitude holding $J$ and $K$.

Let $\operatorname {Long}_J$ and $\operatorname {Long}_K$ denote the longitude of $J$ and $K$ respectively, measured in degrees.

Let $\operatorname {Lat}_J$ denote the latitude of $J$ (and $K$).

Then:

$JK \approx 60 \times \size {\operatorname {Long}_J - \operatorname {Long}_K} \cos \operatorname {Lat}_J$ nautical miles

Proof

We have that $\size {\operatorname {Long}_J - \operatorname {Long}_K}$ is the magnitude of the spherical angle between the meridians on which $J$ and $K$ lie.

Thus from Length of Arc of Small Circle:

$JK \approx \size {\operatorname {Long}_J - \operatorname {Long}_K} \cos \operatorname {Lat}_J$ degrees of arc

Then by definition of nautical mile:

$1$ nautical mile $= \dfrac 1 {60}$ degree of arc of latitude along a meridian measured along Earth's surface.

Thus, approximately, $1$ degree of arc along the equator equals $60$ nautical miles.

The result follows.

$\blacksquare$

Sources

• 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $4$. Terrestrial latitude and longitude.