Find Goh Calculator

Line of Sight Height Calculator

Calculate the minimum height an observer needs to be to see an object at a certain distance, considering the Earth’s curvature. This is useful for line-of-sight calculations.

Example Required Observer Heights (GOH)

Distance (D)	Object Height (ho=10m)	Object Height (ho=50m)
10 km	0.00 m	0.00 m
20 km	2.15 m	0.00 m
30 km	12.16 m	0.96 m
50 km	59.04 m	25.89 m
70 km	133.78 m	78.68 m
100 km	307.72 m	204.31 m

Table showing calculated Required Observer Height (GOH) in meters for different distances and two object heights (10m and 50m), using Earth Radius 6371km.

What is a Required Observer Height (GOH) Calculator?

A Required Observer Height (GOH) Calculator, often related to line of sight over the horizon, is a tool used to determine the minimum height an observer needs to be at to see an object at a specified distance, considering the curvature of the Earth. When looking at objects far away, the Earth’s roundness can hide them below the horizon. The calculator helps find the “Ground Observer Height” or more accurately, the required observer height above the ground or sea level to establish a line of sight to the top of the distant object.

This calculator is essential for anyone dealing with long-distance observation, telecommunications (like microwave links), navigation, surveying, and even amateur radio. If you need to see a tower, a ship, or another point from afar, this calculator helps determine if it’s possible from your current height or how high you need to go. Our Required Observer Height Calculator provides quick and accurate estimations based on geometric optics over a spherical Earth.

Common misconceptions include thinking the Earth is flat for line-of-sight calculations (which is incorrect for long distances) or ignoring the object’s own height, which significantly affects visibility. The Required Observer Height Calculator accounts for both the object’s height and Earth’s curvature.

Required Observer Height (GOH) Formula and Mathematical Explanation

The calculation is based on the geometry of a circle (representing the Earth) and tangents from the observer and the object to the Earth’s surface at the horizon point(s) between them.

Let:

• R be the radius of the Earth.
• h_o be the height of the object above the surface.
• h_g (GOH) be the height of the observer above the surface.
• D be the great circle distance between the base of the observer and the base of the object. For line-of-sight, we approximate it as the sum of distances to the horizon.

The distance to the horizon from a height ‘h’ is given by d = √((R+h)² – R²) = √(2Rh + h²).
If the observer at h_g can just see the top of the object at h_o at distance D, then D is the sum of their distances to their respective horizon points (assuming a single tangent point if they are at the limit of visibility over the curve):
D ≈ √(2Rh_g + h_g²) + √(2Rh_o + h_o²)

To find h_g (GOH), we rearrange:

√(2Rh_g + h_g²) = D – √(2Rh_o + h_o²)

Let K = D – √(2Rh_o + h_o²). If K ≤ 0, it means the object is visible from h_g = 0 or below, so the required height is 0.

If K > 0, we square both sides:

2Rh_g + h_g² = K²

h_g² + 2Rh_g – K² = 0

Solving this quadratic equation for h_g (and taking the positive root as height cannot be negative):

h_g = -R + √(R² + K²)

Our Required Observer Height Calculator uses this formula.

Variable	Meaning	Unit	Typical Value/Range
D	Distance to object	km	1 – 500 km
ho	Height of object	m	0 – 1000 m
R	Earth’s radius	km	6357 – 6378 (avg. 6371) km
K	Intermediate distance	km	0 – D km
hg (GOH)	Required Observer Height	m	≥ 0 m

Variables used in the Required Observer Height Calculator.

Practical Examples (Real-World Use Cases)

Let’s see how the Required Observer Height Calculator works with some examples.

Example 1: Seeing a boat mast

You are on the shore and want to see a boat with a mast 15 meters high, which is 30 km away. Earth’s radius is 6371 km.

• D = 30 km
• h_o = 15 m
• R = 6371 km

Using the Required Observer Height Calculator, the required observer height (GOH) is approximately 8.52 meters. You would need to be on a cliff or building at least 8.52 meters high to see the top of the mast.

Example 2: Radio link between two towers

Two radio towers are 60 km apart. One tower (object) is 50 meters high. What is the minimum height the antenna on the other tower (observer) needs to be for a direct line of sight, assuming smooth Earth?

• D = 60 km
• h_o = 50 m
• R = 6371 km

The Required Observer Height Calculator shows the GOH needed is about 52.01 meters. The second tower’s antenna needs to be at least 52.01 meters high.

How to Use This Required Observer Height Calculator

1. Enter Distance to Object (D): Input the distance in kilometers from you to the object you wish to see.
2. Enter Height of Object (h_o): Input the height of the object in meters above the ground or sea level.
3. Enter Earth’s Radius (R): The average radius (6371 km) is pre-filled, but you can adjust it for local conditions or higher precision.
4. Calculate: The calculator updates in real-time, or you can click “Calculate GOH”.
5. Read Results: The “Required Observer Height (GOH)” is the main result, shown in meters. Intermediate values used in the calculation are also displayed. If GOH is 0 m, the object is visible from ground level at that distance.
6. Use Chart and Table: The chart and table provide visual aids to understand how GOH changes with distance and object height.

Understanding the results helps in planning observations, setting up communication links, or assessing visibility over long distances. Our Required Observer Height Calculator is a powerful tool for these tasks.

Key Factors That Affect Required Observer Height Results

• Distance to Object (D): The further the object, the greater the Earth’s curvature effect, and thus a higher observer height is needed. GOH increases roughly with the square of the distance after accounting for object height.
• Height of Object (h_o): Taller objects can be seen from further away or from lower observer heights. Increasing object height reduces the required GOH.
• Earth’s Radius (R): While relatively constant, using a more accurate local radius can slightly affect results. The standard 6371 km is an average.
• Atmospheric Refraction: The atmosphere bends light (refracts it), especially near the surface. Standard refraction effectively increases the Earth’s radius by about 1/6th (k-factor of 4/3), allowing one to see slightly further or from a lower height than the geometric calculation suggests. Our basic Required Observer Height Calculator does not include refraction by default, but you can simulate it by using a larger effective Earth radius (e.g., 6371 * 4/3 ≈ 8495 km).
• Terrain and Obstacles: The calculator assumes a smooth spherical Earth between the observer and the object. Hills, buildings, or other terrain will block the line of sight and require a greater observer height than calculated.
• Observer’s and Object’s Base Elevation: The calculator assumes both are at the same base elevation (e.g., sea level). If they are at different base elevations, those need to be factored in relative to a common datum.

Frequently Asked Questions (FAQ)

What does GOH stand for?

In this context, we interpret GOH as “Ground Observer Height,” referring to the height above the ground or reference surface required for an observer. The calculator finds this required height.

Does this calculator account for atmospheric refraction?

No, the basic calculation is purely geometric. To approximate standard refraction (k=4/3), you can enter an effective Earth radius of about 8495 km instead of 6371 km in the Required Observer Height Calculator.

What if the calculated GOH is 0 meters?

A GOH of 0 meters means the object is visible from ground level (or the base elevation) at the given distance due to its own height.

Can I use this for very long distances, like hundreds of kilometers?

Yes, the formula is valid, but for very long distances, atmospheric conditions and refraction become much more significant and variable, making the geometric calculation less precise. Also, physical line of sight is usually limited before extreme distances.

What if there are hills between me and the object?

This Required Observer Height Calculator assumes a smooth Earth. Hills or other obstructions need to be considered separately; you’d need to clear the highest obstruction along the path.

How accurate is the Earth’s radius of 6371 km?

It’s a good average (mean radius). The Earth is slightly flattened, with an equatorial radius of about 6378 km and a polar radius of about 6357 km. For very precise calculations, use the radius relevant to your latitude and direction of observation.

Why does the chart show different lines?

The chart illustrates how the required observer height changes with distance for two different object heights (10m and 50m), allowing you to compare the effect of object height.

Can I calculate the maximum distance I can see an object from a certain height?

Yes, you can rearrange the formula or use the Required Observer Height Calculator iteratively. If your height is fixed (h_g) and you know the object height (h_o), the maximum distance D would be √(2Rh_g + h_g²) + √(2Rh_o + h_o²).