The calculated distance (air line) between Buxton and Stoke-on-Trent is approximately 21 miles respectively 34 kilometers.
By car or train, the actual journey to Stoke-on-Trent is certainly longer, as only the direct route (as the crow flies) between Buxton and Stoke-on-Trent has been calculated here.
Air line (approximately)
34 kilometers
18 nautical miles