Re: Could you see from San Francisco to Reno Nevada?

What a fantastic question to discuss on a car trip! Let me see if I can help you out.

First, we need to know what the shortest distance is between San Francisco and Reno, NV. There is a great website that allows you to enter two cities anywhere in the world and calculate the distance between them “as the crow flies”. It is run by an Indonesian Tourism company, but works for cities worldwide (see link below). This website calculated that it is 191 miles from San Francisco, CA, to Reno, NV.

Next, we need to incorporate the curvature of the Earth. Since the Earth is a sphere, it makes sense that when we look out in a straight line, the surface of the Earth actually curves away below our line-of-sight. Math Central at the University of Regina in Saskatchewan, Canada, has solved this problem using some algebra and the Pythagorean Theorem. You can find their initial solution to the problem of how much the Earth curves away in one mile at the link below (answer: the Earth curves about 8 inches in the first mile). However, the problem is slightly more complicated, because the farther you look away, the more the Earth has curved out below your line of sight. Math Central gives a more general mathematical solution for this problem at the second link below. But let me show you a graph where I’ve plotted up the solution and then we can discuss your question by looking at the graph.

What I’ve plotted in this graph is the general mathematical solution to the question: “How high does a building have to be in order to see it from a certain distance away?” I’ve assumed that both the person looking at the building and the base of the building are at sea level (so I’ve used the radius of the Earth – 3963 miles). I’ve calculated everything in miles. To read this graph, simply choose the “Distance away” that you want to see, and then look up the graph to the red line. Read over to the left side of the graph at the level of the red line to see how many miles tall a building would have to be for the top of it to be visible above your horizon. If you think about this graph in another way, it also tells you how far the Earth has curved out from below your line of site for a given distance.

For your example of San Francisco to Reno, NV, I’ve drawn in a vertical blue line at 191 miles (the shortest distance between the two cities). Look at where the blue line intersects the red line and read over to the “Height” side of the graph to see how high a building in Reno would have to be in order to see it from San Francisco. That building in Reno would have to be over 4.5 (four and a half) miles tall! That is very tall and far taller than the tallest building in the world (Taipei 101 in Taipei, Taiwan is only 1671 feet, or barely a third of a mile, tall). In fact, looking at the graph again, you would have to be fewer than 50 miles away from Taipei 101 in order to see the tip of it above your horizon (of course, you would need a telescope and at that distance, the Earth’s atmosphere be too thick to see through clearly, so there are other problems that would become important).

That makes the simple answer to your question “no." You would not be able to see any buildings in Reno, NV, from San Francisco. The Earth is a sphere and its surface curves out from underneath you too quickly. Now, because this is an interesting question and I was curious, I took the problem one step further and included the fact that San Francisco and Reno, NV, are not located at sea level. From the same Indonesian website I mentioned above, I found out that San Francisco is an average of 63 feet above sea level while Reno, NV, is 4498 feet above sea level. Including the relative elevations (but still ignoring any objects in between San Francisco and Reno that might get in the way of us seeing Reno), we could see any building that is over 3.75 miles tall. Unfortunately, that building does not exist either. Ah well…

I hope this helps to answer your question. It’s a fascinating question and I had a lot of fun playing around and figuring out the answer! Thanks for asking!

Calculate the distance between any two cities in the world: http://www.indo.com/distance/

Math Central Web Pages:
How much does the Earth curve away in one mile? http://mathcentral.uregina.ca/QQ/database/QQ.09.97/dyck2.html

A more complete solution to the algebraic problem of Earth’s curvature: http://mathcentral.uregina.ca/QQ/database/QQ.09.02/shirley3.html