[You can consider an n-sphere as a sum of (n-1)-spheres.](https://math.stackexchange.com/a/2903950/466748). An n-sphere can be constructed by stacking scaled (n-1)-spheres.

[![n+1 ball as sum of n-balls][1]][1]

And you can turn around that process. 

The n-sphere has n+1 coordinates and if you take n of the coordinates and scale them to get the appropriate length of the desired radius of the sphere, then you have the n coordinates of a n-1 sphere.

  [1]: https://i.sstatic.net/EcNZJ.png