My sister called me yesterday with a question she dreamed up on the long drive to work. Because I stink at math I’m going to put it to You, Dear Reader. Anyone have an answer?

If there were a perfectly straight road going all the way around the world, and you could see forever, there would be a point where you could not see the road anymore because it curved over the horizon.

If you were to make a bridge as long as the Earth’s diameter, with the center resting on the surface of the Earth, and the ends not flexing at all, what would be the altitude of the ends? Assume the Earth is a perfect sphere with no mountains. Here’s a picture:

A wise man knows when to say, “I don’t know.”

I agree. She has far too long a drive! Maybe you should buy her some audio books for her birthday to entertain her.

Well, I believe you’d have a square between the end of the bridge in question, its tangent to the surface, and the center of the earth. Sides would be the length of the radius of earth: 12742/2=6371km each. Then we find the length of the diagonal of the square (from the point in question to the center of the earth) as square-root-of-two (1.4142) * 6371km= 9010km. From that, subtract the radius and you get 9010km-6371km = 2639km

OK, the distance from the center of the earth to that corner is r * square root of two, cause it’s the hypotenuse of an equilateral right triangle. The distance from the center of the earth to the surface of the earth is r by definition. So it’s r*(sqrt 2) – r, that is, it’s (sqrt 2)-1 times r, and going with 12,742km as the diameter, r is 6371km, so the answer is 2639km.

pretty easy to do using squares and triangles.

The line intersecting two points, 90 degrees apart on the earth’s surface should be 9009.95km long. Recognize that you can form a square within the circle of the earth, each side being this length. Two sides of that square form triangles with each side of your bridge. Because they are right-angle triangles, the distance from the 90 degree corner to the hypotenuse will be exactly half the hypotenuse. That gets us close. Now, if we subtract our 9009.95 figure (one side of the square) from 12742 (the diameter of the earth), and half the result, we have the difference between the distance from the angle to the square and the distance from the angle to the earth. Simple subtraction then gives us a figure of 2638.95km distance from the bridge end to the surface of the earth at the nearest point.

Well, your sister is also a definate product of your parents.~~~smile again.

I tried to figure this out but I got a headache, so let’s just say Tim and Dave are in agreement and I would really like to meet their parents. Kathy, I agree with you and I already know you have caring parents that are humorous.