Earth Curvature Bulge Calculator

The calculator

Enter a distance below, choose your units and press calculate to find the height of the bulge across the distance due to the curvature of the earth.

The mathematics

We want to work out the height h, for a given distance d.

To calculate this we need to first work out the distance l through the earth to the center point .

We'll need this angle θ in degrees first of all.

Given that d is a fraction of the earth's diameter r, then θ is simply calculated with the following equation.

Once we have θ, we can work out l as so.

The radius of the earth r and l form a right angle triangle, so using half of θ we can work out l with the following bit of trig, that you will have most likely learned in school.

Now that we know l, calculating h is simple using pythagoras.

As you can see we simply need to calculate f and subtract it from r to find h, the following formula will do that. See if you can spot the reverse pythagoras you also may have learned in school.

And there you have it, quite simple maths when broken down to its base components, yet understandable by any secondary level student the world over. The whole thing can be simplified down to the following.

To be honest I'd use the calculator. It's easier, and if you check the javascript you can see it doing all the same maths as above, albeit with radians instead of degrees.

Don't take my word for it, check my maths, I could be wrong. Trust no one.