In reading a clinical trial, I came across the following: This study was powered to show a 10% difference in disease-free survival (from 75% to 85), corresponding to a hazard ratio of 0.56.

Can someone explain how the hazard ratio is calculated from the absolute difference/survival probabilities in this scenario?

As @pmbrown mentioned, under the presumption that the distribution of event times are exponential, you can get the hazard rate for each event free survival proportion (presuming time t = 12 months), using R:

-(log(0.75)) / 12
[1] 0.02397351

-(log(0.85)) / 12
[1] 0.01354324

The hazard ratio is then:

0.01354324 / 0.02397351
[1] 0.5649252

Essentially, use the same units (e.g. in months) for the time t value in both of the first two lines to get the respective hazard rate for each and then divide to get the ratio.

If you then want to get the estimated median survival for each (again in months):

-(log(0.5)) / 0.02397351
[1] 28.91305

-(log(0.5)) / 0.01354324
[1] 51.18031

Note that:

28.91305 / 51.18031
[1] 0.5649253

The CRAB/SWOG folks have a conversion tool for these various units at: