Image by Dominic.
Turns out you can calculate the distance to the horizon, or the distance to the dream yacht, by knowing your height and the Earth's radius. More precisely, it is not your height but rather the height of your eyes measured from the water’s surface that is of interest here. Denote it as h and assume it is around 6 feet. Denote R as the Earth's radius, which is approximately 3,959 miles = 20,903,520 feet. The drawing below shows you standing on the Earth’s surface with your eyes being h+R distance away from the Earth's center, your dream yacht on the horizon, on a distance R from Earth's center, and distance d away from you. The triangle between these three points has a right angle near the horizon.
All that separates you from the dream yacht is this distance d. Can you swim it? Bet you never thought Pythagoras’ Theorem would come in handy. The equations may make your head spin, but we are talking matters of critical importance here. So, let's tackle it. All it says is that in a right triangle (triangle with a 90 degree angle) the square of it's longest side is equal to the sum of squares of the shorter sides.
(R+h )2 = R2 + d2
R2 + 2Rh + h2 = R2 + d2
d2 = 2Rh + h2
d = √(2Rh) (because h2 is very small compared with the Earth's radius, we discard it)
Using our numbers, R= 20,903,520 feet and h=6 feet, we get:
d is √( 2 x 20,903,520 x 6) in feet.
d is 15838 feet.
d is 3 miles.
Your dream yacht with linen-white sails is 3 miles away from you.
Even with these numbers being approximate and many assumptions made along the way, swimming 3 miles in the champaign bubbles of sun rays may be rather dangerous. And if you are one of those people who already have a yacht and dream of sinking and being saved by an attractive Baywatch lifeguard (or Michael Phelps) – let's hope they have a motor boat.
See another math mystery from The Math Mom: what your closet has in common with National Enquirer magazine and Homeland Security alerts.