how would I go about finding the distance between two points on earth

Question

I want to try and find the distance between two points on earth, as much by hand as possible. Could anyone give me ideas on how I would go about doing this, (explanations as deep as possible please!)

Answer 1

Assume that earth is a perfect sphere with radius $R$. Let $p_1, p_2$ be the points of interest, laying on the surface of the earth (i.e. $||p_i|| = R$). The points $p_1, p_2$, seen as vectors, create an angle $\theta$ between them. Thus, the distance (shortest path on the surface from $p_1$ to $p_2$) is given by $\theta R$. Draw a picture to convince yourself. The angle is given by $$\theta = \arccos(\frac{p_1 \cdot p_2}{R^2})$$