Crofton's Theorem can be proven by showing that the area of the set of poles of equators that intersect an arc on the Equator of the unit sphere equals four times the length of the arc. If we approximate a curve on the sphere by a spherical polygon, then the integral of the number of times an equator intersects the polygon will give four times the length of the polygon, which is Crofton's formula when we go to the limit of the inscribed spherical polygons.