# If two people are facing toward each other how far apart could they both see a 500 ft tall tower?

Assume two people are facing each other on a level plain and there is 500 foot tall tower between them. What is the maximum distance they could be separated from each other and still keep the tower in view?

This is the classic “Distance to the Horizon” calculation.
This website says each person would be 27.2 miles from the tower, so the total distance would be twice that, or 54.4 miles.

ETA: Oh, I just say you said “level plain.” In that case, it’s infinitely far…

Curvature of the earth is approximately 8 inches per mile so you could see the top of the tower until you were 500 ft/8 inches = 750 miles away from it. With one person on each side the distance between them would be 1500 miles. This assumes excellent eyesight. 55 miles. I first added the observer’s five feet above ground level to the tower’s 500 feet, on the theory that seeing from five feet high to 500 feet high was the same as seeing from 505 feet high to the ground. Now using this distance-to-horizon calculator, I get 27.5 miles. Doubling that gives 55 miles.

On reflection my answer doesn’t really pass the sanity check. It appears that I did not read far enough into my own reference. I defer to beowulff.

This is clearly incorrect. Have you never driven before? One can’t even see mountains that far away.

I think you need to subtract the person’s height from the height of the tower…

In any practical application of this, would the visual acuity of the people be the limiting factor?

Clearly not, that would mean a taller person would have to be closer to see the tower than a smaller person which is obviously not true. This doesn’t give the same answer because it’s an approximation.

If you don’t have the handy-dandy distance to horizon calculator this should work. Call the radius of the earth (or other planet) r and the height of the object h. Then the top of the object is r+h from the center of the earth. The line form the object’s top to the horizon is tangent to the earth. This tangency point is r from the center of the earth. The tangent line and the line from the tangency point to the center of the earth form a right angle. Therefore the angle a between radius to the object and the tangency point is a = arccos(r/(r+h)). If you express a in radians then the distance measured along the great circle of the earth between the base of the object and the tangency point is is C*(a/2pi) where C is the circumference of the earth. Or since C = 2pir, the distance is ra.

For this problem you have to solve this twice – once for the tower and once for the person, then add them and double.

How wide is the tower?

and how reflective and is there a light on top, etc?