Start at one point on a circle and go, along the circle, to the antipode, the farthest point on the circle from where you started. You travel a semicircle. Then consider the same endpoints, but take a shortcut through the center of the circle. You travel the diameter. The ratio between the length of the route staying on the circle versus going straight through is pi/2 ~= 1.570796327, which we'll call the semicircle constant. (2pi/4 if you like 2pi, but 4 is more complicated than 2 in the denominator, and division by 2 matches well with the concept of a semicircle, so pi is better than 2pi in this situation.)

This constant of proportionality remains the same comparing routes on versus through spheres, and (I think) for all higher dimensional hyperspheres.

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