When navigating in areas where there is a large tidal range, the height of tide in relation to MHWS must be taken into account if the charted height of the object is being used in navigational calculations. The height of objects shown on navigational charts is the height above the level for ‘mean high water springs’ (MHWS) which is used as the chart datum for heights. Summary of formula for calculating the distance that an object will appear on the horizon taking into account height of eye:ĭ = √(12734.9 x h 1/1000) + √ (12734.9 x h 2/1000) Km. So we can see that taking the observer’s height of eye into account makes quite a difference to the result. Using the formula that we previously established for finding the distance from the top of an object to the horizon, we have: To calculate the value of AB, we must first calculate the values of AT and BT. H 2 represents the height of the lighthouse (120m.). H 1 represents the observer’s height of eye (say 4m.) So the height of eye must obviously be taken into account and the problem now becomes as shown in the following diagram: Of course, the height of the observer’s eye will be several metres above sea level (say 3 – 4 m. The above calculation was made with the assumption that the observer’s eye is at sea level. Summary of Formulas for finding the distance from the top of an object to the horizon, : Therefore, assuming the observer is at sea level, the top of the lighthouse would become visible at a distance of 39.09 Km. So the answer to the above example in nautical miles becomes: 39.09 x 0.54 = 21.1 n.m. (In fact, it is easier to multiply by 0.54 which is the inverse of 1.85). so, if we want to convert the answer to nautical miles, we must divide by 1.85. To convert to nautical miles: 1 nautical mile = 1.85 Km. we can put these values into the formula √ (2rh) as follows:ĭistance =√ (2×6367.45 x 120/1000) (divide h by 1000 to convert to Km.) and the height of the Pharos light was 120 m. Since we know that the mean radius of the Earth is 6367.45 Km. Since the heights of objects marked on modern navigational charts are usually given in metres, it would be easier to calculate the distance in kilometres instead of nautical miles. In the following example, we will imagine that the Pharos Lighthouse (ht. Calculating the distance to an object on the horizon. So the formula for calculating the distance from the top of an object to the horizon becomes: √ (2rh)Įxample. h 2 can be discounted since r is many, many times greater than h). Pythagoras provides us with a method of solving this problem as shown below:
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