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Distance by vertical sextant angle – measuring range with a sextant

Measure the vertical angle of an object of known height – a lighthouse, a church tower or a mountain – with a sextant, and you can work out its distance. Together with a bearing, that gives you a fix.

• Vertical angle
• Object height
• Base visible / hidden
• Height of eye
• Distance in nm

What is the vertical angle?

The vertical angle is the angle at which you see an object from its base to its top. The closer you get, the larger the angle. Because the height is known, the angle gives the distance.

There are two cases:

• Base visible: You can see the object down to the waterline. The angle is measured from the waterline to the top.
• Base beyond the horizon: The object is so far away that its lower part is hidden behind the horizon – typical for mountains. You then measure from the horizon to the top, and your height of eye comes into it.

When do you need it?

• Fix from bearing and distance: A bearing and a vertical angle of the same object give a position – without radar and without a second object.
• Keeping clear: Off a coast with shoals you can set a minimum distance and check continuously with the vertical angle of a lighthouse that you stay far enough off – the classic vertical danger angle.
• Making a landfall by day: Mountain tops show up long before the coast. Their vertical angle gives an idea of the distance from far away.

Formulas and working

Base visible

• Distance in nautical miles
• H – object height in metres
• α – vertical angle in minutes of arc (1° = 60′)

Why 13/7? For small angles the distance equals the height divided by the angle in radians. One minute of arc is 1/3438 radian and a nautical mile is 1852 m. 3438 / 1852 gives 1.857 – almost exactly 13/7. With the height in feet the factor becomes about 0.566.

Base beyond the horizon

Here the height of eye h comes in, because the angle is measured above the horizon:

1. Auxiliary value

   Vertical angle minus dip: A = α − 1.8 · √h

2. Root term

   W = 3.71 · (H − h) + A²

3. Distance

   Distance = √W − A

Worked examples

Things to watch out for

• Index error: Always correct the sextant reading for index error first. With small angles, a single minute of arc makes a big difference.
• The right height: The object height must match the angle you measured. Measuring from the waterline, you need the height of the top above the water, not above the ground. For lights, the chart gives the elevation of the light, not the height of the tower.
• Tides: Chart heights refer to a high water level. At low water the object is a little higher than charted – noticeable at short distances.
• Large angle, short distance: The simple formula is meant for small angles. At a few cables and large angles it becomes less accurate.

How it works in NauticTools2

1. Enter the vertical angle

   Degrees and minutes, already corrected for index error.

2. Enter the object height

   In metres, from the chart or the list of lights.

3. Base visible or not

   If the base is hidden, untick the box and enter the height of eye – it is filled in from the settings.

4. Calculate

   NauticTools2 shows the distance and the full working. Inputs can be saved, loaded and printed.

Frequently asked questions

Where does the factor 13/7 come from?

From converting minutes of arc to radians and metres to nautical miles: 3438 / 1852 ≈ 1.857 ≈ 13/7.

Do dip and refraction matter?

Hardly, when the base is visible, because both ends of the angle are on the object. With the base hidden you measure against the horizon – which is why the dip (1.8 · √h) is part of the extended formula.

Can you do it without a sextant?

Roughly, for example with binoculars with a reticle. For useful distances, though, a sextant is far more accurate.

Related tools

• Geographical Range of Lights – distance to a light just rising on the horizon
• Index Error – find the sextant's index error
• Bearing Conversion – convert the bearing of the object to a true bearing