How to measure the world
(r + h)2 = r2 + L2
Expand the left-hand term to give:
r2 + 2rh + h2 = r2 + L2
Cancel the r2 terms to give
2rh + h2 = L2
Use the fact that h . LESS THAN r to neglect h LESS THAN
2rh = L2
Rearrange in terms of r:
r = L22h
Throughout history, people have climbed mountains to learn things - whether to meditate on the wonder of creation and their own insignificance therein or just to look out for enemy armies. With a little maths, you and your class can also calculate the size of the Earth.
The Ancients reasoned that the world is roughly spherical because the shadow cast on the Moon during a lunar eclipse was always circular. The next question was obvious: how big is it?
Many impressive methods were devised to calculate this dimension; so many that a separate branch of maths arose. Geometry literally means "Earth measurement".
Eratosthenes compared the length of midsummer noon shadows at two places - the Tropic of Cancer, with the Sun overhead and so no shadow, and 600 miles to the north, with a shadow angle of 7.5 degrees. He then "scaled up" the 600 miles to estimate the Earth's circumference.
Another method used the time it takes a camel train to disappear over the desert horizon. Given the height and speed of a camel, one could estimate the Earth's curvature and thus radius.
We will use the same principle but, in the absence of a desert, climb a hill instead. Imagine it's a clear day and we climb a hill - a conveniently isolated hill above a low-lying plain. We saunter up, free from the burden of theodolites, sextants or aneroid barometers. We just need good eyes and a map.
The Ordnance Survey has kindly marked the height of our hill on the map, saving a lot of trigonometry. So all we need to do is catch our breath and stare into the distance. After consulting the map, and deciding whether that distant city is Sheffield or Manchester (or is it Venice?), you should have some idea how far you can see.
Remember - other hills don't count because they could be showing over the "natural" horizon.
In the diagram, the essential insight is that the line of sight AB is tangential to the Earth's diameter and thus perpendicular to the radius BC.
Wherever we see a right-angled triangle, Pythagoras applies.
Take a real example, remembering to work in consistent units, because if h is in metres, r in miles and L in kilometres, we'll get into all sorts of confusion. The best choice is probably kilometres.
Scafell Pike is 977m, or 0.977km high. On a clear day, you can see about 110km. From our formula, we would estimate r as around 6,200 km.
That's not far from the generally accepted dimension of 6,370 km. How close can you get?