Circumference of Circle on Curved (Sphere) Surface

The circumference of a circle drawn on a sphere of radius R is R sin(r/R) where r is the great circle distance along the sphere from the center of the circle to its edge. This 2-D example is followed for the 3-D positively curved hypersphere, in which the circumference of a circle is less than the Euclidean value of 2r.

The circle-circumference relation for circles of ever larger radius on a sphere of radius R. Again, this mirrors exactly the relation for circle circumferences within a 3-D hypersphere. The relation is: C = R sin(r/R) . In this example, the value of R is 4000 miles, to create a 3-D hyperspherical volume with curvature radius equal to that of the Earth.

Figure: this website.