How to Calculate Distance to Horizon and Line of Sight

My first couple of questions after calculating curvature was well how far can I see? and how far does a building or object have to be before I can't see it anymore? So we need to figure out where the horizon is. More math! Yay!

You can download the calculator I made on my other post here.

Calculating Distance to Horizon

To calculate this we use the Pythagorean Theorem again but just solving for the distance instead of height. If you want the height equation click here.

So we start with this

If a² + b² = c² then for this example we get radius² + distance² = (height + radius)². So to solve for distance we just subtract radius from both sides and square the whole thing. Then we get distance = √(r + h)² - r².

If we fill in the numbers we get distance = √(3963 + .001136)² - 3963²   (To get miles we divide 6 feet by 5280 which equals .001136). This then becomes distance = √9.00393 = 3.00. So a person with their eye level at 6 feet will see the horizon at 3 miles.

If you wanted to use an approx. method just use 1.22 *√h.

Line of Sight

So now we can use the same equation to find out how far away we can see an object. First you add the distance to horizon for each height together.  So if there are 2 people with an eye level of 6 feet we know they can each see 3 miles. 3 + 3 = 6. So this means they can see each other at a Maximum of 6 miles.

So if we know how far any height can be from the horizon we can determine how much is visible over the curve of the earth. We can take the Maximum distance and subtract the distance from object 1 to object 2. Then use the equation √(r² + d²) - r = h or approximate formula h = (d/1.22)².

So let's try an example. If a person has an eye level of 6 feet and they are viewing a building 6 miles away. The building is 30 feet tall. We know a 6 foot eye level can see 3 miles to the horizon. Then we calculate and distance for a 30 foot building to be 6.7 miles. So the total possible distance for a person to see the tip of the building is 9.7 miles. Then subtract the distance between the person and the object. 9.7 - 6 = 3.7 miles. 

Now we can plug that in. √(3963² + 3.7²) - 3963² =  .001727. Multiply that by 5280 we get 9.12 feet. Using the approx. method (3.7/1.22)² = 9.2 feet.  Using either method we find that about 9 feet of the building is hidden below the horizon.

If you need to account for refraction there is an approximate method for that as well. Instead of 1.22 use 1.32. So the approximate refraction formula would be h = (d/1.32)².

If I didn't explain it well enough, Wikipedia does a great job here. 

Please do not hesitate to let me know if I made any mistakes. I would like all the information to be correct so any feedback is appreciated! Thank you!