The Earth is round… duh!

Around 500 BC Pythagoras and his Greek mates noticed that during a Moon eclipse, the shadow of the Earth on the Moon was always round. If the Earth was a disk, its shadow would look like an oval most of the time. The only shape whose shadow is always round, no matter where the light comes from, is a sphere. They concluded that the Earth shape was a sphere. Smart!

The earth roundness limits how far I can see from the seaside. Walking along the beach near Byron I wondered about that. My eyes are 1.7m from the ground, I call this height h. If R is the Earth radius, the distance between them and the Earth centre is R+h. If I call d the distance to the horizon, I can apply Pythagoras rule to the triangle formed by my eyes, the Earth centre and a point anywhere on the horizon:

R² + d² = (R+h)²

The Earth radius R is about 6371km. I spare you the maths. With d and h expressed in metres, the solution is:

d = 3569 x (square root of h)

So if my eyes are at h=1.7 m from the ground, the horizon is at 3569 x 1.304 = 4,653 m, about 4.6 km.

Let’s go up to the top of the Byron lighthouse to broaden our horizon. The light sits at 118 m from sea level. The horizon from up there grows to 38.77 km. No wonder they insist on building light houses on top of hills!

References:

Pythagoras: https://en.wikipedia.org/wiki/Pythagoras#In_astronomy

Byron lighthouse: https://lighthouses.org.au/nsw/cape-byron-lighthouse/