## (8a) Distance to the HorizonImagine you were standing at an elevation ofh meters above the ocean and looking out across the water. What is the distance D to the horizon? It can be calculated, if you know the radius R of the Earth.
Your line of sight to the horizon is a It follows that the triangle OAB obeys the theorem of Pythagoras, which here can be written (OA) or if the length of each line is spelled out (R + h) By an algebraic identity (derived in the "mathematical refresher"), the left-hand side equals R R If now h(2R + h) = D The diameter 2Rh = D where SQRT stands here for "square root of". This equation lets one calculate D--in kilometers, if SQRT(2Rh) = (SQRT(2R))(SQRT(h)) with the two square roots multiplied. Using D = 112.88 km SQRT(h) If you are standing atop a mountain 1 km high, The calculation should also hold the other way around. From a boat on the ocean you should begin seeing the top of Mauna Kea after you pass a distance of 226 km (again, not accounting for refraction). On November 15, 1806, Lieutenant Zebulon Pike of the US Army, leading an exploration party across the plains of the midwestern US, saw through his spyglass the top of a distant peak, just above the horizon. It took his party a week to cover the 100 miles to the mountain, which is now known as Pike's Peak, one of the tallest in Colorado. Pike actually tried to climb to its top, but the snow and the unexpected height of the mountain forced him back. |

Sites on Pike's Peak here and here.

**Further Exploring: #8b How distant is the Moon?**

**Next Regular Stop: #9 Discovery of the Solar System**

Author and curator: David P. Stern, u5dps@lepvax.gsfc.nasa.gov

Last updated 24 August 1998