# Mathematics: Sextant angles

## Vertical sextant angle

The triangle OBL can be described in terms of H, **α** and Distance:

**Distance = H / tan( α)**

where, the angle is in **radians** (0—2**π**) and both height and distance are in metres.

- The relations between radians and degrees is:

**α**= A ×**π**/ 180

with “A” being the same angle in degrees. - To describe angle A in minutes total, then A × 60 = a, thus
**α**= (a / 60) × (**π**/ 180). So,**α**= a / 3438, “a” being the angle in arc minutes. *Fact*: tan(x) = x, if angle x is small.

Resulting in (with **π** = 3.14): **Distance **(m)**= H × 3438 / a**

- Furthermore, distance in Nautical Miles = distance in meters / 1852.

Voilà, a very practical equation:

**Distance = 1.856 × H / a**

It contains just two approximations, both of neglitible influence. First, we left out the *tan* function and second we used 3.14 for **π**.

Please realize that a smaller angle improves the approximation of the *tan*.
Yet, as an opposing effect the instrument error of a smaller sextant angle increases.

All in all, the factor 1.856 is *not* a typo, and just by chance near to the nautical mile: 1.852 kilometres.
If you are still reading, you are very brave person and might agree that it originates from (60 × 180) / (**π** × 1852).

So far we considered a perfect triangle (OBL) and forgot that life isn't always perfect. Height h is usually quite small, but distance SB sometimes is not.
This leads to an extra premise, which is seldom mentioned by other navigation textbooks:

**Angle OLS should be bigger than 15°**.

Further reading:

Without tides the Mediterranean is the perfect cruising venue with
**RYA & ASA sailing schools** out of **Athens** into the Saronic or Cyclades (Santorini, Paros, Mykonos,…) and in the Ionian: **Lefkas**,

or back to **chapter 5 of the navigation course**,

or **learn to sail bareboat** with a **private RYA ASA instructor**.