Geometry at the Beach

Tim and I were in Rehoboth this last weekend, and had a hotel room that looked out over the ocean. We were talking about ships coming over the horizon, and how far out to sea one would be able to spot a ship. The more general question became, how far away is the horizon?

Tim put on his mathematics cap, and I got out my trusty pen and paper. Mind you, Tim did all of this without drawing any pretty pictures, which makes me think he keeps all of his geometry in his head.

Here's the fact I needed to know: the earth's radius is approximately 3950 miles. It's a little less at the poles, a little more at the equator. I was standing about 40 feet above the ground on the hotel balcony. I left the hotel maid a very cryptic scrap of paper:

horizon distance = (((r + h)^2) - (r^2))^(1/2)

Where:
r = the earth's radius: 3950 miles x 5280 ft/1 mile = 20,856,000 ft
h = observation height above the ground = 40 ft

I bet you didn't think that the handy distance formula you learned in high school geometry would ever have any use at the beach! Pythagorus didn't have Arabic numerals, and a handy calculator. I'll leave it up to the reader to do the math, but it comes out to about 7.74 miles, which is just about what I thought it was when I eyeballed it. I should have been a surveyor