I propose Haversine formula for exposition of material reg. distances on earth.

Vincenty’s formula calculates distances to millimeter order. Bowring and Lambert’s formula are easier than Vincenty’s but do not have good accuracy especially if distances on earth are quite large.

You should know spherical earth formula, all the other formula mentioned here are oblate-sphere formula. For flat earth, where you can use your ordinary Euclidean rules such as $\sin^2\theta + \cos^2 \theta=1$, you can just project distances on earth to a plane but the results you will get may have undesirable errors because first of all earth is a large sphere and your approximations will be valid only for short distances because you are all “frog in the well” you learned about the well to measure the earth.

When the frog went out of the well it met Riemann and he told that he had been having some real-time fun which Euclid did not. The frog came back and told this to all others and they all went out to the town bought over Riemanian formula books and started learning about the earth.

When the frogs met Vincenty, he told he knows how to compute distances on earth surface to accuracy of millimeters but one needs a great deal of computational power and many number of iterations.

What a cool site I got in this respect, you can plug in my latitude, longitude, and you can get x and distances, what a cool … ;